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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Common_pitfalls"> <div class="vector-toc-text"> 3 Common pitfalls </div> </a> <ul id="toc-Common_pitfalls-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Performance_issues" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Performance_issues"> <div class="vector-toc-text"> 4 Performance issues </div> </a> <ul id="toc-Performance_issues-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_(identities)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties_(identities)"> <div class="vector-toc-text"> 5 Properties (identities) </div> </a> <ul id="toc-Properties_(identities)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_programming_languages" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_programming_languages"> <div class="vector-toc-text"> 6 In programming languages </div> </a> <ul id="toc-In_programming_languages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 7 Generalizations </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations subsection </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Modulo_with_offset" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modulo_with_offset"> <div class="vector-toc-text"> 7.1 Modulo with offset </div> </a> <ul id="toc-Modulo_with_offset-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implementing_other_modulo_definitions_using_truncation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Implementing_other_modulo_definitions_using_truncation"> <div class="vector-toc-text"> 7.2 Implementing other modulo definitions using truncation </div> </a> <ul id="toc-Implementing_other_modulo_definitions_using_truncation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 10 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 11 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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Available in 26 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-26" aria-hidden="true" > 26 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%A7%D9%85%D9%84_%D9%82%D8%B3%D9%85%D8%A9_%D9%85%D8%B9_%D8%A8%D8%A7%D9%82%D9%8A" title="عامل قسمة مع باقي – Arabic" lang="ar" hreflang="ar" data-title="عامل قسمة مع باقي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Operaci%C3%B3_m%C3%B2dul" title="Operació mòdul – Catalan" lang="ca" hreflang="ca" data-title="Operació mòdul" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Modulo" title="Modulo – Danish" lang="da" hreflang="da" data-title="Modulo" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Division_mit_Rest#Modulo" title="Division mit Rest – German" lang="de" hreflang="de" data-title="Division mit Rest" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Operaci%C3%B3n_m%C3%B3dulo" title="Operación módulo – Spanish" lang="es" hreflang="es" data-title="Operación módulo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%DB%8C%D8%A7%D8%AA_%D9%BE%DB%8C%D9%85%D8%A7%D9%86%D9%87" title="عملیات پیمانه – Persian" lang="fa" hreflang="fa" data-title="عملیات پیمانه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Modulo_(op%C3%A9ration)" title="Modulo (opération) – French" lang="fr" hreflang="fr" data-title="Modulo (opération)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AA%A8%EB%93%88%EB%A1%9C" title="모듈로 – Korean" lang="ko" hreflang="ko" data-title="모듈로" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%B8%D5%A4%D5%B8%D6%82%D5%AC%D5%B8_%D5%A3%D5%B8%D6%80%D5%AE%D5%B8%D5%B2%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Մոդուլո գործողություն – Armenian" lang="hy" hreflang="hy" data-title="Մոդուլո գործողություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Operasi_modulus" title="Operasi modulus – Indonesian" lang="id" hreflang="id" data-title="Operasi modulus" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/M%C3%B3dular" title="Módular – Icelandic" lang="is" hreflang="is" data-title="Módular" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Operazione_modulo" title="Operazione modulo – Italian" lang="it" hreflang="it" data-title="Operazione modulo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/MOD_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%86%D0%B8%D1%8F%D1%81%D1%8B" title="MOD операциясы – Kazakh" lang="kk" hreflang="kk" data-title="MOD операциясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%89%B0%E4%BD%99%E6%BC%94%E7%AE%97" title="剰余演算 – Japanese" lang="ja" hreflang="ja" data-title="剰余演算" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Modulus" title="Modulus – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Modulus" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Modulo" title="Modulo – Polish" lang="pl" hreflang="pl" data-title="Modulo" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Opera%C3%A7%C3%A3o_m%C3%B3dulo" title="Operação módulo – Portuguese" lang="pt" hreflang="pt" data-title="Operação módulo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Modulo" title="Modulo – Romanian" lang="ro" hreflang="ro" data-title="Modulo" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Modulation_(mathematics)" title="Modulation (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Modulation (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Modulo" title="Modulo – Slovenian" lang="sl" hreflang="sl" data-title="Modulo" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B3%D1%80%D1%83%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0" title="Конгруенција – Serbian" lang="sr" hreflang="sr" data-title="Конгруенција" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv badge-Q70893996 mw-list-item" title=""><a href="https://sv.wikipedia.org/wiki/Modulor%C3%A4kning" title="Moduloräkning – Swedish" lang="sv" hreflang="sv" data-title="Moduloräkning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Modulo_i%C5%9Flemi" title="Modulo işlemi – Turkish" lang="tr" hreflang="tr" data-title="Modulo işlemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_to%C3%A1n_modulo" title="Phép toán modulo – Vietnamese" lang="vi" hreflang="vi" data-title="Phép toán modulo" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%95%86%E9%A4%98" title="商餘 – Cantonese" lang="yue" hreflang="yue" data-title="商餘" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A8%A1%E9%99%A4" title="模除 – Chinese" lang="zh" hreflang="zh" data-title="模除" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Modulo_operator&redirect=no" class="mw-redirect" title="Modulo operator">Modulo operator</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the binary operation mod(a,n). For the (mod n) notation, see <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a>. For other uses, see <a href="/wiki/Modulo_(disambiguation)" class="mw-disambig" title="Modulo (disambiguation)">Modulo (disambiguation)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Computational operation</div> In <a href="/wiki/Computing" title="Computing">computing</a>, the modulo operation returns the <a href="/wiki/Remainder" title="Remainder">remainder</a> or signed remainder of a <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, after one number is divided by another, called the <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulus</a> of the operation. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> of a by n, where a is the <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">dividend</a> and n is the <a href="/wiki/Divisor" title="Divisor">divisor</a>.<a href="#cite_note-1">[1]</a> For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a <a href="/wiki/Quotient" title="Quotient">quotient</a> of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being <a href="/wiki/Integer" title="Integer">integers</a>, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of n is 0 to n − 1. a mod 1 is always 0. When exactly one of a or n is negative, the basic definition breaks down, and <a href="/wiki/Programming_languages" class="mw-redirect" title="Programming languages">programming languages</a> differ in how these values are defined. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Variants_of_the_definition">Variants of the definition</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=1" title="Edit section: Variants of the definition">edit</a>]</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the result of the <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> operation is an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a>, and any member of the class may be chosen as <a href="/wiki/Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">representative</a>; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a>).<a href="#cite_note-2">[2]</a> However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the <a href="/wiki/Programming_language" title="Programming language">programming language</a> or the underlying <a href="/wiki/Computer_hardware" title="Computer hardware">hardware</a>. In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy the following conditions: <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 3.2em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&q\in \mathbb {Z} \\&a=nq+r\\&|r|<|n|\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>a</mi> <mo>=</mo> <mi>n</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&q\in \mathbb {Z} \\&a=nq+r\\&|r|<|n|\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385ef8ce0954e0cf4067ba7b29b11bd829a0e280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:11.433ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}&q\in \mathbb {Z} \\&a=nq+r\\&|r|<|n|\end{aligned}}}"></td> <td></td> <td class="nowrap">1</td></tr></tbody></table> This still leaves a sign ambiguity if the remainder is non-zero: two possible choices for the remainder occur, one negative and the other positive; that choice determines which of the two consecutive quotients must be used to satisfy equation (1). In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of a or n.<a href="#cite_note-3">[a]</a> Standard <a href="/wiki/Pascal_(programming_language)" title="Pascal (programming language)">Pascal</a> and <a href="/wiki/ALGOL_68" title="ALGOL 68">ALGOL 68</a>, for example, give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it to the implementation when either of n or a is negative (see the table under <a href="#In_programming_languages">§ In programming languages</a> for details). Some systems leave a modulo 0 undefined, though others define it as a. <div><ul><li><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Divmod_truncated.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Divmod_truncated.svg/260px-Divmod_truncated.svg.png" decoding="async" width="260" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Divmod_truncated.svg/390px-Divmod_truncated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Divmod_truncated.svg/520px-Divmod_truncated.svg.png 2x" data-file-width="411" data-file-height="213" /></a><figcaption><style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style>  Quotient (q) and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959">  remainder (r) as functions of dividend (a), using truncated division</figcaption></figure> Many implementations use truncated division, for which the quotient is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\operatorname {trunc} \left({\frac {a}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>trunc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\operatorname {trunc} \left({\frac {a}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5feb729ea32f718e78b3a8e49dacb1ca391f2bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.608ex; height:4.843ex;" alt="{\displaystyle q=\operatorname {trunc} \left({\frac {a}{n}}\right)}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {trunc} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>trunc</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {trunc} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69473479d8f591cfd14c480ab725aac0c2229d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.434ex; height:2.009ex;" alt="{\displaystyle \operatorname {trunc} }"> is the <a href="/wiki/Integral_part" class="mw-redirect" title="Integral part">integral part function</a> (<a href="/wiki/Rounding#Rounding_toward_zero" title="Rounding">rounding toward zero</a>), i.e. the <a href="/wiki/Truncation" title="Truncation">truncation</a> to zero significant digits. Thus according to equation (<a href="#math_1">1</a>), the remainder has the same sign as the dividend <var style="padding-right: 1px;">a</var> so can take 2|n| − 1 values: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a-n\operatorname {trunc} \left({\frac {a}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mi>trunc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a-n\operatorname {trunc} \left({\frac {a}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/883e082a5405c785bd79385c89571b70e88f7fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.439ex; height:4.843ex;" alt="{\displaystyle r=a-n\operatorname {trunc} \left({\frac {a}{n}}\right)}"></dd></dl></li><li><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Divmod_floored.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Divmod_floored.svg/260px-Divmod_floored.svg.png" decoding="async" width="260" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Divmod_floored.svg/390px-Divmod_floored.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Divmod_floored.svg/520px-Divmod_floored.svg.png 2x" data-file-width="411" data-file-height="213" /></a><figcaption>Quotient and remainder using floored division</figcaption></figure> <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a><a href="#cite_note-4">[3]</a> promotes floored division, for which the quotient is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\left\lfloor {\frac {a}{n}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\left\lfloor {\frac {a}{n}}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19bd75a99c368ba756c7c1037ddc5531f7580775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.854ex; height:4.843ex;" alt="{\displaystyle q=\left\lfloor {\frac {a}{n}}\right\rfloor }"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \,\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \,\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3084289cca849fb18ec6407b9de5faa5846d8607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.452ex; height:2.843ex;" alt="{\displaystyle \lfloor \,\rfloor }"> is the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a> (<a href="/wiki/Rounding#Rounding_down" title="Rounding">rounding down</a>). Thus according to equation (<a href="#math_1">1</a>), the remainder has the same sign as the divisor <var style="padding-right: 1px;">n</var>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a-n\left\lfloor {\frac {a}{n}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a-n\left\lfloor {\frac {a}{n}}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1ad1c9a2d7af2202b14284c0262355bad5cdb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.685ex; height:4.843ex;" alt="{\displaystyle r=a-n\left\lfloor {\frac {a}{n}}\right\rfloor }"></dd></dl></li><li><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Divmod_Euclidean.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Divmod_Euclidean.svg/260px-Divmod_Euclidean.svg.png" decoding="async" width="260" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Divmod_Euclidean.svg/390px-Divmod_Euclidean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Divmod_Euclidean.svg/520px-Divmod_Euclidean.svg.png 2x" data-file-width="411" data-file-height="213" /></a><figcaption>Quotient and remainder using Euclidean division</figcaption></figure> Raymond T. Boute<a href="#cite_note-5">[4]</a> promotes <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a>, for which the quotient is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\operatorname {sgn}(n)\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor ={\begin{cases}\left\lfloor {\frac {a}{n}}\right\rfloor &{\text{if }}n>0\\\left\lceil {\frac {a}{n}}\right\rceil &{\text{if }}n<0\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mo>|</mo> <mi>n</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\operatorname {sgn}(n)\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor ={\begin{cases}\left\lfloor {\frac {a}{n}}\right\rfloor &{\text{if }}n>0\\\left\lceil {\frac {a}{n}}\right\rceil &{\text{if }}n<0\\\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee5d36b5d3186ddd85c92de8ad4e410e1f469a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.893ex; height:6.509ex;" alt="{\displaystyle q=\operatorname {sgn}(n)\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor ={\begin{cases}\left\lfloor {\frac {a}{n}}\right\rfloor &{\text{if }}n>0\\\left\lceil {\frac {a}{n}}\right\rceil &{\text{if }}n<0\\\end{cases}}}"></dd></dl> where sgn is the <a href="/wiki/Sign_function" title="Sign function">sign function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \,\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor \,\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3084289cca849fb18ec6407b9de5faa5846d8607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.452ex; height:2.843ex;" alt="{\displaystyle \lfloor \,\rfloor }"> is the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a> (<a href="/wiki/Rounding#Rounding_down" title="Rounding">rounding down</a>), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lceil \,\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌈</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⌉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lceil \,\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2a7e2cf4cbfa5a2507240f4d3f15350c171d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.452ex; height:2.843ex;" alt="{\displaystyle \lceil \,\rceil }"> is the <a href="/wiki/Ceiling_function" class="mw-redirect" title="Ceiling function">ceiling function</a> (<a href="/wiki/Rounding#Rounding_up" title="Rounding">rounding up</a>). Thus according to equation (<a href="#math_1">1</a>), the remainder is non negative: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a-|n|\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mo>|</mo> <mi>n</mi> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a-|n|\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ab7a5aa1dac63cd7fa2b5a41095fdcd2500bf3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.528ex; height:6.343ex;" alt="{\displaystyle r=a-|n|\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor }"></dd></dl></li><li><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Divmod_rounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Divmod_rounding.svg/260px-Divmod_rounding.svg.png" decoding="async" width="260" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Divmod_rounding.svg/390px-Divmod_rounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Divmod_rounding.svg/520px-Divmod_rounding.svg.png 2x" data-file-width="411" data-file-height="213" /></a><figcaption>Quotient and remainder using rounded division</figcaption></figure> Common Lisp and <a href="/wiki/IEEE_754-1985" title="IEEE 754-1985">IEEE 754</a> use rounded division, for which the quotient is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\operatorname {round} \left({\frac {a}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>round</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\operatorname {round} \left({\frac {a}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f001dd60b59e4f44c5c44a2f2853dc33a8015a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.126ex; height:4.843ex;" alt="{\displaystyle q=\operatorname {round} \left({\frac {a}{n}}\right)}"></dd></dl> where round is the <a href="/wiki/Rounding" title="Rounding">round function</a> (<a href="/wiki/Rounding#Rounding_half_to_even" title="Rounding">rounding half to even</a>). Thus according to equation (<a href="#math_1">1</a>), the remainder falls between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {n}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {n}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b28ab851ef83cb88352dc199323826d4095709c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.039ex; height:4.676ex;" alt="{\displaystyle -{\frac {n}{2}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1216d48de276dc45542cb80b1e49037131ec9624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.231ex; height:4.676ex;" alt="{\displaystyle {\frac {n}{2}}}">, and its sign depends on which side of zero it falls to be within these boundaries: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a-n\operatorname {round} \left({\frac {a}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mi>round</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a-n\operatorname {round} \left({\frac {a}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5229d5b5d726332e30acd9ebb9c98823cf43f10d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.957ex; height:4.843ex;" alt="{\displaystyle r=a-n\operatorname {round} \left({\frac {a}{n}}\right)}"></dd></dl></li><li><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Divmod_ceiling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Divmod_ceiling.svg/260px-Divmod_ceiling.svg.png" decoding="async" width="260" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Divmod_ceiling.svg/390px-Divmod_ceiling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Divmod_ceiling.svg/520px-Divmod_ceiling.svg.png 2x" data-file-width="411" data-file-height="213" /></a><figcaption>Quotient and remainder using ceiling division</figcaption></figure> Common Lisp also uses ceiling division, for which the quotient is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\left\lceil {\frac {a}{n}}\right\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\left\lceil {\frac {a}{n}}\right\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a842cab18be465dda3798f42dd382e3057859c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.854ex; height:4.843ex;" alt="{\displaystyle q=\left\lceil {\frac {a}{n}}\right\rceil }"></dd></dl> where ⌈⌉ is the <a href="/wiki/Ceiling_function" class="mw-redirect" title="Ceiling function">ceiling function</a> (<a href="/wiki/Rounding#Rounding_up" title="Rounding">rounding up</a>). Thus according to equation (<a href="#math_1">1</a>), the remainder has the opposite sign of that of the divisor: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a-n\left\lceil {\frac {a}{n}}\right\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>n</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a-n\left\lceil {\frac {a}{n}}\right\rceil }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11bd7c338a1cee1ab548508404dc460df2b7077d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.685ex; height:4.843ex;" alt="{\displaystyle r=a-n\left\lceil {\frac {a}{n}}\right\rceil }"></dd></dl></li></ul></div> If both the dividend and divisor are positive, then the truncated, floored, and Euclidean definitions agree. If the dividend is positive and the divisor is negative, then the truncated and Euclidean definitions agree. If the dividend is negative and the divisor is positive, then the floored and Euclidean definitions agree. If both the dividend and divisor are negative, then the truncated and floored definitions agree. As described by Leijen, <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions.<div class="templatequotecite">— <cite>Daan Leijen, Division and Modulus for Computer Scientists<a href="#cite_note-Leijen-6">[5]</a></cite></div></blockquote> However, truncated division satisfies the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({-a})/b={-(a/b)}=a/({-b})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>b</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({-a})/b={-(a/b)}=a/({-b})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c17b403df86515893e20dc9a2eb128516de8312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.219ex; height:2.843ex;" alt="{\displaystyle ({-a})/b={-(a/b)}=a/({-b})}">.<a href="#cite_note-7">[6]</a> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=2" title="Edit section: Notation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">This section is about the binary mod operation. For the (mod m) notation, see <a href="/wiki/Congruence_relation" title="Congruence relation">congruence relation</a>.</div> Some calculators have a mod() function button, and many programming languages have a similar function, expressed as mod(a, n), for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder <a href="/wiki/Operator_(programming)" class="mw-redirect" title="Operator (programming)">operator</a>, such as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">a % n</code> or <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">a mod n</code>. For environments lacking a similar function, any of the three definitions above can be used. <div class="mw-heading mw-heading2"><h2 id="Common_pitfalls">Common pitfalls</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=3" title="Edit section: Common pitfalls">edit</a>]</div> When the result of a modulo operation has the sign of the dividend (truncated definition), it can lead to surprising mistakes. For example, to test if an integer is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a>, one might be inclined to test if the remainder by 2 is equal to 1: <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>bool is_odd(int n) { return n % 2 == 1; } </pre></div> But in a language where modulo has the sign of the dividend, that is incorrect, because when n (the dividend) is negative and odd, n mod 2 returns −1, and the function returns false. One correct alternative is to test that the remainder is not 0 (because remainder 0 is the same regardless of the signs): <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>bool is_odd(int n) { return n % 2 != 0; } </pre></div> Or with the binary arithmetic: <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>bool is_odd(int n) { return n & 1; } </pre></div> <div class="mw-heading mw-heading2"><h2 id="Performance_issues">Performance issues</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=4" title="Edit section: Performance issues">edit</a>]</div> Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of <a href="/wiki/Powers_of_2" class="mw-redirect" title="Powers of 2">powers of 2</a> can alternatively be expressed as a <a href="/wiki/Bitwise_operation" title="Bitwise operation">bitwise</a> AND operation (assuming x is a positive integer, or using a non-truncating definition): <dl><dd><code>x % 2n == x & (2n - 1)</code></dd></dl> Examples: <dl><dd><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">x % 2 == x & 1</code></dd> <dd><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">x % 4 == x & 3</code></dd> <dd><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">x % 8 == x & 7</code></dd></dl> In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.<a href="#cite_note-8">[7]</a> <a href="/wiki/Compiler_optimization" class="mw-redirect" title="Compiler optimization">Compiler optimizations</a> may recognize expressions of the form <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">expression % constant</code> where <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">constant</code> is a power of two and automatically implement them as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">expression & (constant-1)</code>, allowing the programmer to write clearer code without compromising performance. This simple optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend (including <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a>), unless the dividend is of an <a href="/wiki/Signedness" title="Signedness">unsigned</a> integer type. This is because, if the dividend is negative, the modulo will be negative, whereas <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">expression & (constant-1)</code> will always be positive. For these languages, the equivalence <code>x % 2n == x < 0 ? x | ~(2n - 1) : x & (2n - 1)</code> has to be used instead, expressed using bitwise OR, NOT and AND operations. Optimizations for general constant-modulus operations also exist by calculating the division first using the <a href="/wiki/Division_algorithm#Division_by_a_constant" title="Division algorithm">constant-divisor optimization</a>. <div class="mw-heading mw-heading2"><h2 id="Properties_(identities)">Properties (identities)</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=5" title="Edit section: Properties (identities)">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Modular_arithmetic#Properties" title="Modular arithmetic">Modular arithmetic § Properties</a></div> Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> proofs, such as the <a href="/wiki/Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">Diffie–Hellman key exchange</a>. The properties involving multiplication, division, and exponentiation generally require that a and n are integers. <ul><li>Identity: <ul><li>(a mod n) mod n = a mod n.</li> <li>nx mod n = 0 for all positive integer values of x.</li> <li>If p is a <a href="/wiki/Prime_number" title="Prime number">prime number</a> which is not a <a href="/wiki/Divisor" title="Divisor">divisor</a> of b, then abp−1 mod p = a mod p, due to <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a>.</li></ul></li> <li>Inverse: <ul><li>[(−a mod n) + (a mod n)] mod n = 0.</li> <li>b−1 mod n denotes the <a href="/wiki/Modular_multiplicative_inverse" title="Modular multiplicative inverse">modular multiplicative inverse</a>, which is defined <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> b and n are <a href="/wiki/Relatively_prime" class="mw-redirect" title="Relatively prime">relatively prime</a>, which is the case when the left hand side is defined: [(b−1 mod n)(b mod n)] mod n = 1.</li></ul></li> <li>Distributive: <ul><li>(a + b) mod n = [(a mod n) + (b mod n)] mod n.</li> <li>ab mod n = [(a mod n)(b mod n)] mod n.</li></ul></li> <li>Division (definition): <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠a/b⁠ mod n = [(a mod n)(b−1 mod n)] mod n, when the right hand side is defined (that is when b and n are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>), and undefined otherwise.</li> <li>Inverse multiplication: [(ab mod n)(b−1 mod n)] mod n = a mod n.</li></ul> <div class="mw-heading mw-heading2"><h2 id="In_programming_languages">In programming languages</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=6" title="Edit section: In programming languages">edit</a>]</div> <table class="wikitable sortable"> <caption>Modulo operators in various programming languages </caption> <tbody><tr> <th scope="col"><a href="/wiki/Programming_language" title="Programming language">Language</a> </th> <th scope="col">Operator </th> <th scope="col">Integer </th> <th scope="col">Floating-point </th> <th abbr="Sign" scope="col"><a href="#Variants_of_the_definition">Definition</a> </th></tr> <tr> <td><a href="/wiki/ABAP" title="ABAP">ABAP</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Euclidean </td></tr> <tr> <td><a href="/wiki/ActionScript" title="ActionScript">ActionScript</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Ada_(programming_language)" title="Ada (programming language)">Ada</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-Ada-9">[8]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-Ada-9">[8]</a> </td></tr> <tr> <td><a href="/wiki/ALGOL_68" title="ALGOL 68">ALGOL 68</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">÷×</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean </td></tr> <tr> <td><a href="/wiki/AMPL" title="AMPL">AMPL</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/APL_(programming_language)" title="APL (programming language)">APL</a> </td> <td><code>|</code><a href="#cite_note-rev-10">[b]</a> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/AppleScript" title="AppleScript">AppleScript</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/AutoLISP" title="AutoLISP">AutoLISP</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">(rem d n)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/AWK" title="AWK">AWK</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Bash_(Unix_shell)" title="Bash (Unix shell)">bash</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/BASIC" title="BASIC">BASIC</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Varies by implementation </td></tr> <tr> <td><a href="/wiki/Bc_(programming_language)" title="Bc (programming language)">bc</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="3"><a href="/wiki/C_(programming_language)" title="C (programming language)">C</a> <a href="/wiki/C%2B%2B" title="C++">C++</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">div</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-c-13">[c]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fmod</code> (C) <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">std::fmod</code> (C++) </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-14">[11]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder</code> (C) <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">std::remainder</code> (C++) </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Rounded </td></tr> <tr> <td rowspan="2"><a href="/wiki/C_Sharp_(programming_language)" title="C Sharp (programming language)">C#</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Math.IEEERemainder</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Rounded<a href="#cite_note-.NET-15">[12]</a> </td></tr> <tr> <td><a href="/wiki/Clarion_(programming_language)" title="Clarion (programming language)">Clarion</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Clean_(programming_language)" title="Clean (programming language)">Clean</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Clojure" title="Clojure">Clojure</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-16">[13]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-17">[14]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/COBOL" title="COBOL">COBOL</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">FUNCTION MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-isocobol-18">[15]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">FUNCTION REM</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-isocobol-18">[15]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/CoffeeScript" title="CoffeeScript">CoffeeScript</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-CoffeeScript-19">[16]</a> </td></tr> <tr> <td><a href="/wiki/ColdFusion" class="mw-redirect" title="ColdFusion">ColdFusion</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Common_Intermediate_Language" title="Common Intermediate Language">Common Intermediate Language</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> (signed) </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-20">[17]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem.un</code> (unsigned) </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">— </td></tr> <tr> <td rowspan="2"><a href="/wiki/Common_Lisp" title="Common Lisp">Common Lisp</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Crystal_(programming_language)" title="Crystal (programming language)">Crystal</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modulo</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/CSS" title="CSS">CSS</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored<a href="#cite_note-21">[18]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-22">[19]</a> </td></tr> <tr> <td><a href="/wiki/D_(programming_language)" title="D (programming language)">D</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-23">[20]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Dart_(programming_language)" title="Dart (programming language)">Dart</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Euclidean<a href="#cite_note-24">[21]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-25">[22]</a> </td></tr> <tr> <td><a href="/wiki/Eiffel_(programming_language)" title="Eiffel (programming language)">Eiffel</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">\\</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Elixir_(programming_language)" title="Elixir (programming language)">Elixir</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem/2</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-26">[23]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Integer.mod/2</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-27">[24]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Elm_(programming_language)" title="Elm (programming language)">Elm</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modBy</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-28">[25]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainderBy</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-29">[26]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Erlang_(programming_language)" title="Erlang (programming language)">Erlang</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">math:fmod/2</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated (same as C)<a href="#cite_note-30">[27]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Euphoria_(programming_language)" title="Euphoria (programming language)">Euphoria</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/F_Sharp_(programming_language)" title="F Sharp (programming language)">F#</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Math.IEEERemainder</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Rounded<a href="#cite_note-.NET-15">[12]</a> </td></tr> <tr> <td><a href="/wiki/Factor_(programming_language)" title="Factor (programming language)">Factor</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/FileMaker" title="FileMaker">FileMaker</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="3"><a href="/wiki/Forth_(programming_language)" title="Forth (programming language)">Forth</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Implementation defined </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fm/mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">sm/rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Fortran" title="Fortran">Fortran</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modulo</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Frink_(programming_language)" title="Frink (programming language)">Frink</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="2"><a href="/wiki/Full_BASIC" title="Full BASIC">Full BASIC</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored<a href="#cite_note-31">[28]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">REMAINDER</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-32">[29]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/GLSL" class="mw-redirect" title="GLSL">GLSL</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Undefined<a href="#cite_note-33">[30]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored<a href="#cite_note-34">[31]</a> </td></tr> <tr> <td><a href="/wiki/GameMaker_Studio" class="mw-redirect" title="GameMaker Studio">GameMaker Studio</a> (GML) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="4"><a href="/wiki/Godot_(game_engine)" title="Godot (game engine)">GDScript (Godot)</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fmod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">posmod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fposmod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Euclidean </td></tr> <tr> <td rowspan="4"><a href="/wiki/Go_(programming_language)" title="Go (programming language)">Go</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-35">[32]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">math.Mod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-36">[33]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">big.Int.Mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean<a href="#cite_note-37">[34]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">big.Int.Rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-38">[35]</a> </td></tr> <tr> <td><a href="/wiki/Apache_Groovy" title="Apache Groovy">Groovy</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="3"><a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-Haskell_2010-39">[36]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-Haskell_2010-39">[36]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Data.Fixed.mod'</code> (<a href="/wiki/Glasgow_Haskell_Compiler" title="Glasgow Haskell Compiler">GHC</a>) </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Haxe" title="Haxe">Haxe</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/HLSL" class="mw-redirect" title="HLSL">HLSL</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Undefined<a href="#cite_note-40">[37]</a> </td></tr> <tr> <td><a href="/wiki/J_(programming_language)" title="J (programming language)">J</a> </td> <td><code>|</code><a href="#cite_note-rev-10">[b]</a> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="2"><a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Math.floorMod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/JavaScript" title="JavaScript">JavaScript</a> <a href="/wiki/TypeScript" title="TypeScript">TypeScript</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored<a href="#cite_note-41">[38]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-42">[39]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Kotlin_(programming_language)" title="Kotlin (programming language)">Kotlin</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-43">[40]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored<a href="#cite_note-44">[41]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/KornShell" title="KornShell">ksh</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated (same as POSIX sh) </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fmod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/LabVIEW" title="LabVIEW">LabVIEW</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/LibreOffice" title="LibreOffice">LibreOffice</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">=MOD()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="2"><a href="/wiki/Logo_(programming_language)" title="Logo (programming language)">Logo</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MODULO</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">REMAINDER</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Lua_(programming_language)" title="Lua (programming language)">Lua</a> 5 </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Lua_(programming_language)" title="Lua (programming language)">Lua</a> 4 </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod(x,y)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Liberty_BASIC" title="Liberty BASIC">Liberty BASIC</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Mathcad" title="Mathcad">Mathcad</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod(x,y)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="3"><a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">e mod m</code> (by default), <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modp(e, m)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mods(e, m)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Rounded </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">frem(e, m)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Rounded </td></tr> <tr> <td><a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Mod[a, b]</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="2"><a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Maxima_(software)" title="Maxima (software)">Maxima</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Maya_Embedded_Language" title="Maya Embedded Language">Maya Embedded Language</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">=MOD()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Minitab" title="Minitab">Minitab</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="2"><a href="/wiki/Modula-2" title="Modula-2">Modula-2</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">REM</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/MUMPS" title="MUMPS">MUMPS</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">#</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="2"><a href="/wiki/Netwide_Assembler" title="Netwide Assembler">Netwide Assembler</a> (<a href="/wiki/Netwide_Assembler" title="Netwide Assembler">NASM</a>, <a href="/wiki/Netwide_Assembler" title="Netwide Assembler">NASMX</a>) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">div</code> (unsigned) </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">— </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%%</code> (signed) </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Implementation-defined<a href="#cite_note-45">[42]</a> </td></tr> <tr> <td><a href="/wiki/Nim_(programming_language)" title="Nim (programming language)">Nim</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Oberon_(programming_language)" title="Oberon (programming language)">Oberon</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored-like<a href="#cite_note-46">[d]</a> </td></tr> <tr> <td><a href="/wiki/Objective-C" title="Objective-C">Objective-C</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated (same as C99) </td></tr> <tr> <td><a href="/wiki/Object_Pascal" title="Object Pascal">Object Pascal</a>, <a href="/wiki/Delphi_(programming_language)" class="mw-redirect" title="Delphi (programming language)">Delphi</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/OCaml" title="OCaml">OCaml</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-47">[43]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod_float</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-48">[44]</a> </td></tr> <tr> <td><a href="/wiki/Occam_(programming_language)" title="Occam (programming language)">Occam</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">\</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Pascal_(programming_language)" title="Pascal (programming language)">Pascal</a> (ISO-7185 and -10206) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean-like<a href="#cite_note-49">[e]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Perl" title="Perl">Perl</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored<a href="#cite_note-51">[f]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">POSIX::fmod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/w/index.php?title=Phix&action=edit&redlink=1" class="new" title="Phix (page does not exist)">Phix</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/PHP" title="PHP">PHP</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-52">[46]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fmod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-53">[47]</a> </td></tr> <tr> <td>PIC <a href="/wiki/BASIC" title="BASIC">BASIC</a> Pro </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">\\</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/PL/I" title="PL/I">PL/I</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored (ANSI PL/I) </td></tr> <tr> <td><a href="/wiki/PowerShell" title="PowerShell">PowerShell</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td>Programming Code (<a href="/wiki/PRC_(Palm_OS)" title="PRC (Palm OS)">PRC</a>) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MATH.OP - 'MOD; (\)'</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Undefined </td></tr> <tr> <td><a href="/wiki/OpenEdge_Advanced_Business_Language" title="OpenEdge Advanced Business Language">Progress</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modulo</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Prolog" title="Prolog">Prolog</a> (<a rel="nofollow" class="external text" href="https://www.iso.org/standard/21413.html">ISO 1995</a>) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/PureBasic" title="PureBasic">PureBasic</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Mod(x,y)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/PureScript" title="PureScript">PureScript</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">`mod`</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean<a href="#cite_note-purescript_euclid-54">[48]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Pure_Data" title="Pure Data">Pure Data</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated (same as C) </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td rowspan="3"><a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">math.fmod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">math.remainder</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Rounded </td></tr> <tr> <td><a href="/wiki/Q_Sharp" title="Q Sharp">Q#</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-55">[49]</a> </td></tr> <tr> <td><a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored<a href="#cite_note-56">[50]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Racket_(programming_language)" title="Racket (programming language)">Racket</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modulo</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Raku_(programming_language)" title="Raku (programming language)">Raku</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/RealBasic" class="mw-redirect" title="RealBasic">RealBasic</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Reason_(programming_language)" title="Reason (programming language)">Reason</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Rexx" title="Rexx">Rexx</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">//</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/RPG_(programming_language)" class="mw-redirect" title="RPG (programming language)">RPG</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%REM</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modulo()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Rust_(programming_language)" title="Rust (programming language)">Rust</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem_euclid()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Euclidean<a href="#cite_note-rust_rem_euclid-57">[51]</a> </td></tr> <tr> <td><a href="/wiki/SAS_language" title="SAS language">SAS</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">MOD</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Scala_(programming_language)" title="Scala (programming language)">Scala</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/Scheme_(programming_language)" title="Scheme (programming language)">Scheme</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modulo</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="4"><a href="/wiki/Scheme_(programming_language)" title="Scheme (programming language)">Scheme</a> <a href="/wiki/Scheme_(programming_language)#R6RS" title="Scheme (programming language)">R6RS</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean<a href="#cite_note-r6rs-58">[52]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod0</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Rounded<a href="#cite_note-r6rs-58">[52]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">flmod</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Euclidean </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">flmod0</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Rounded </td></tr> <tr> <td><a href="/wiki/Scratch_(programming_language)" title="Scratch (programming language)">Scratch</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td rowspan="2"><a href="/wiki/Seed7" title="Seed7">Seed7</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/SenseTalk" title="SenseTalk">SenseTalk</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">modulo</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/POSIX_shell" class="mw-redirect" title="POSIX shell"><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">sh</code> (POSIX)</a> (includes <a href="/wiki/Bash_(Unix_shell)" title="Bash (Unix shell)">bash</a>, <a href="/wiki/Mksh" class="mw-redirect" title="Mksh">mksh</a>, &c.) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated (same as C)<a href="#cite_note-59">[53]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Smalltalk" title="Smalltalk">Smalltalk</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">\\</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem:</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Snap!_(programming_language)" title="Snap! (programming language)">Snap!</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Parallax_Propeller#Built_in_Spin_byte_code_interpreter" title="Parallax Propeller">Spin</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">//</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Solidity" title="Solidity">Solidity</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-60">[54]</a> </td></tr> <tr> <td><a href="/wiki/SQL" title="SQL">SQL</a> (<a href="/wiki/SQL:1999" title="SQL:1999">SQL:1999</a>) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod(x,y)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/SQL" title="SQL">SQL</a> (<a href="/wiki/SQL:2011" title="SQL:2011">SQL:2011</a>) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="3"><a href="/wiki/Standard_ML" title="Standard ML">Standard ML</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Int.rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Real.rem</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Stata" title="Stata">Stata</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod(x,y)</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean </td></tr> <tr> <td rowspan="3"><a href="/wiki/Swift_(programming_language)" title="Swift (programming language)">Swift</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-61">[55]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">remainder(dividingBy:)</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Rounded<a href="#cite_note-62">[56]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">truncatingRemainder(dividingBy:)</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-63">[57]</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Tcl" title="Tcl">Tcl</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">fmod()</code> </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated (as C) </td></tr> <tr> <td><a href="/wiki/Tcsh" title="Tcsh">tcsh</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Torque_(game_engine)" title="Torque (game engine)">Torque</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Turing_(programming_language)" title="Turing (programming language)">Turing</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Verilog" title="Verilog">Verilog</a> (2001) </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/VHDL" title="VHDL">VHDL</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Floored </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Vimscript" class="mw-redirect" title="Vimscript">VimL</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td><a href="/wiki/Visual_Basic_.NET" class="mw-redirect" title="Visual Basic .NET">Visual Basic</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/WebAssembly" title="WebAssembly">WebAssembly</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">i32.rem_u</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">i64.rem_u</code> (unsigned) </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">—<a href="#cite_note-wasmcs2-64">[58]</a> </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">i32.rem_s</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">i64.rem_s</code> (signed) </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated<a href="#cite_note-wasmcs2-64">[58]</a> </td></tr> <tr> <td><a href="/wiki/X86_assembly_language" title="X86 assembly language">x86 assembly</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">IDIV</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Truncated </td></tr> <tr> <td rowspan="2"><a href="/wiki/XBase%2B%2B" title="XBase++">XBase++</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated </td></tr> <tr> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Mod()</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Floored </td></tr> <tr> <td><a href="/wiki/Zig_(programming_language)" title="Zig (programming language)">Zig</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">@mod</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">@rem</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Truncated<a href="#cite_note-65">[59]</a> </td></tr> <tr> <td><a href="/wiki/Z3_Theorem_Prover" title="Z3 Theorem Prover">Z3 theorem prover</a> </td> <td><code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">div</code>, <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>Euclidean </td></tr></tbody></table> In addition, many computer systems provide a <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">divmod</code> functionality, which produces the quotient and the remainder at the same time. Examples include the <a href="/wiki/X86_architecture" class="mw-redirect" title="X86 architecture">x86 architecture</a>'s <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">IDIV</code> instruction, the C programming language's <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">div()</code> function, and <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>'s <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">divmod()</code> function. <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=7" title="Edit section: Generalizations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Modulo_with_offset">Modulo with offset</h3>[<a href="/w/index.php?title=Modulo&action=edit&section=8" title="Edit section: Modulo with offset">edit</a>]</div> Sometimes it is useful for the result of a modulo n to lie not between 0 and n − 1, but between some number d and d + n − 1. In that case, d is called an offset and d = 1 is particularly common. There does not seem to be a standard notation for this operation, so let us tentatively use a modd n. We thus have the following definition:<a href="#cite_note-Mathematica_Mod-66">[60]</a> x = a modd n just in case d ≤ x ≤ d + n − 1 and x mod n = a mod n. Clearly, the usual modulo operation corresponds to zero offset: a mod n = a mod0 n. The operation of modulo with offset is related to the <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">floor function</a> as follows: <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\operatorname {mod} _{d}n=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msub> <mi>mod</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>d</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\operatorname {mod} _{d}n=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2295c23019fdd9ab8984f08bc8919006923080fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.699ex; height:6.176ex;" alt="{\displaystyle a\operatorname {mod} _{d}n=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor .}"></dd></dl></dd></dl> To see this, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>d</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cecb12b97c06c7fec21b9f083c9922f75b323b3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.579ex; height:4.843ex;" alt="{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor }">. We first show that x mod n = a mod n. It is in general true that (a + bn) mod n = a mod n for all integers b; thus, this is true also in the particular case when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle b=-\!\left\lfloor {\frac {a-d}{n}}\right\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mspace width="negativethinmathspace" /> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>d</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle b=-\!\left\lfloor {\frac {a-d}{n}}\right\rfloor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf3e6789718c0a36c6b0aae5dbe04b93f08f941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.203ex; height:4.843ex;" alt="{\textstyle b=-\!\left\lfloor {\frac {a-d}{n}}\right\rfloor }">; but that means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x{\bmod {n}}=\left(a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor \right)\!{\bmod {n}}=a{\bmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>d</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x{\bmod {n}}=\left(a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor \right)\!{\bmod {n}}=a{\bmod {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55dbae6aaf1928432910a9a06191e870c3cf228c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.911ex; height:4.843ex;" alt="{\textstyle x{\bmod {n}}=\left(a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor \right)\!{\bmod {n}}=a{\bmod {n}}}">, which is what we wanted to prove. It remains to be shown that d ≤ x ≤ d + n − 1. Let k and r be the integers such that a − d = kn + r with 0 ≤ r ≤ n − 1 (see <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a>). Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\lfloor {\frac {a-d}{n}}\right\rfloor =k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>d</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left\lfloor {\frac {a-d}{n}}\right\rfloor =k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7588602c0a11bb9742f4c6edf3c740a1b55dd0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.609ex; height:4.843ex;" alt="{\textstyle \left\lfloor {\frac {a-d}{n}}\right\rfloor =k}">, thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor =a-nk=d+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>d</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>⌋</mo> </mrow> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>n</mi> <mi>k</mi> <mo>=</mo> <mi>d</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor =a-nk=d+r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d428438128efb6d26627bf91adfaeca82b1a743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.557ex; height:4.843ex;" alt="{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor =a-nk=d+r}">. Now take 0 ≤ r ≤ n − 1 and add d to both sides, obtaining d ≤ d + r ≤ d + n − 1. But we've seen that x = d + r, so we are done. The modulo with offset a modd n is implemented in <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">Mod[a, n, d]</code> .<a href="#cite_note-Mathematica_Mod-66">[60]</a> <div class="mw-heading mw-heading3"><h3 id="Implementing_other_modulo_definitions_using_truncation">Implementing other modulo definitions using truncation</h3>[<a href="/w/index.php?title=Modulo&action=edit&section=9" title="Edit section: Implementing other modulo definitions using truncation">edit</a>]</div> Despite the mathematical elegance of Knuth's floored division and Euclidean division, it is generally much more common to find a truncated division-based modulo in programming languages. Leijen provides the following algorithms for calculating the two divisions given a truncated integer division:<a href="#cite_note-Leijen-6">[5]</a> <div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>/* Euclidean and Floored divmod, in the style of C's ldiv() */ typedef struct { /* This structure is part of the C stdlib.h, but is reproduced here for clarity */ long int quot; long int rem; } ldiv_t; /* Euclidean division */ inline ldiv_t ldivE(long numer, long denom) { /* The C99 and C++11 languages define both of these as truncating. */ long q = numer / denom; long r = numer % denom; if (r < 0) { if (denom > 0) { q = q - 1; r = r + denom; } else { q = q + 1; r = r - denom; } } return (ldiv_t){.quot = q, .rem = r}; } /* Floored division */ inline ldiv_t ldivF(long numer, long denom) { long q = numer / denom; long r = numer % denom; if ((r > 0 && denom < 0) || (r < 0 && denom > 0)) { q = q - 1; r = r + denom; } return (ldiv_t){.quot = q, .rem = r}; } </pre></div> For both cases, the remainder can be calculated independently of the quotient, but not vice versa. The operations are combined here to save screen space, as the logical branches are the same. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=10" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Modulo_(disambiguation)" class="mw-disambig" title="Modulo (disambiguation)">Modulo (disambiguation)</a> – many uses of the word modulo, all of which grew out of <a href="/wiki/Carl_F._Gauss" class="mw-redirect" title="Carl F. Gauss">Carl F. Gauss</a>' approach to <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> in 1801.</li> <li><a href="/wiki/Modulo_(mathematics)" title="Modulo (mathematics)">Modulo (mathematics)</a>, general use of the term in mathematics</li> <li><a href="/wiki/Modular_exponentiation" title="Modular exponentiation">Modular exponentiation</a></li> <li><a href="/wiki/Turn_(angle)" title="Turn (angle)">Turn (angle)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=11" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><a href="#cite_ref-3">^</a> Mathematically, these two choices are but two of the infinite number of choices available for <a href="/wiki/Remainder" title="Remainder">the inequality satisfied by a remainder</a>. </li> <li id="cite_note-rev-10">^ <a href="#cite_ref-rev_10-0">a</a> <a href="#cite_ref-rev_10-1">b</a> Argument order reverses, i.e., <code>α|ω</code> computes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega {\bmod {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega {\bmod {\alpha }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43850a0844eb494d3a6100bc0c24e84277614ac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.615ex; height:2.176ex;" alt="{\displaystyle \omega {\bmod {\alpha }}}">, the remainder when dividing <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">ω</code> by <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">α</code>. </li> <li id="cite_note-c-13"><a href="#cite_ref-c_13-0">^</a> <a href="/wiki/C99" title="C99">C99</a> and <a href="/wiki/C%2B%2B11" title="C++11">C++11</a> define the behavior of <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">%</code> to be truncated.<a href="#cite_note-C99-11">[9]</a> The standards before then leave the behavior implementation-defined.<a href="#cite_note-12">[10]</a> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> Divisor must be positive, otherwise undefined. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> As discussed by Boute, ISO Pascal's definitions of <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">div</code> and <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">mod</code> do not obey the Division Identity of D = d · (D / d) + D % d, and are thus fundamentally broken. </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> Perl usually uses arithmetic modulo operator that is machine-independent. For examples and exceptions, see the Perl documentation on multiplicative operators.<a href="#cite_note-50">[45]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Modulo&action=edit&section=12" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Congruence.html">"Congruence"</a>. 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