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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">142857 (número)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir para um artigo noutra língua. Disponível em 26 línguas" > <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" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">26 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/142857" title="142857 — checo" lang="cs" hreflang="cs" data-title="142857" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/142857" title="142857 — dinamarquês" lang="da" hreflang="da" data-title="142857" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/142857" title="142857 — alemão" lang="de" hreflang="de" data-title="142857" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/142857" title="142857 — grego" lang="el" hreflang="el" data-title="142857" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/142857" title="142857 — inglês" lang="en" hreflang="en" data-title="142857" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/142_857" title="142 857 — espanhol" lang="es" hreflang="es" data-title="142 857" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DB%B1%DB%B4%DB%B2%DB%B8%DB%B5%DB%B7" title="۱۴۲۸۵۷ — persa" lang="fa" hreflang="fa" data-title="۱۴۲۸۵۷" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/142_857_(nombre)" title="142 857 (nombre) — francês" lang="fr" hreflang="fr" data-title="142 857 (nombre)" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/142857_(%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE)" title="142857 (संख्या) — hindi" lang="hi" hreflang="hi" data-title="142857 (संख्या)" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/142857_(sz%C3%A1m)" title="142857 (szám) — húngaro" lang="hu" hreflang="hu" data-title="142857 (szám)" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/142.857_(angka)" title="142.857 (angka) — indonésio" lang="id" hreflang="id" data-title="142.857 (angka)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/142857" title="142857 — italiano" lang="it" hreflang="it" data-title="142857" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/142857" title="142857 — japonês" lang="ja" hreflang="ja" data-title="142857" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/142,857" title="142,857 — coreano" lang="ko" hreflang="ko" data-title="142,857" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/142857_(skaitlis)" title="142857 (skaitlis) — letão" lang="lv" hreflang="lv" data-title="142857 (skaitlis)" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/142857" title="142857 — norueguês bokmål" lang="nb" hreflang="nb" data-title="142857" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/142857_(num%C4%83r)" title="142857 (număr) — romeno" lang="ro" hreflang="ro" data-title="142857 (număr)" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/142_857_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="142 857 (число) — russo" lang="ru" hreflang="ru" data-title="142 857 (число)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/142857" title="142857 — sérvio" lang="sr" hreflang="sr" data-title="142857" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/142857_(tal)" title="142857 (tal) — sueco" lang="sv" hreflang="sv" data-title="142857 (tal)" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/142857_(%E0%AE%8E%E0%AE%A3%E0%AF%8D)" title="142857 (எண்) — tâmil" lang="ta" hreflang="ta" data-title="142857 (எண்)" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/142857" title="142857 — tailandês" lang="th" hreflang="th" data-title="142857" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/142857" title="142857 — turco" lang="tr" hreflang="tr" data-title="142857" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/142857_(%D8%B9%D8%AF%D8%AF)" title="142857 (عدد) — urdu" lang="ur" hreflang="ur" data-title="142857 (عدد)" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/142857" title="142857 — vietnamita" lang="vi" hreflang="vi" data-title="142857" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/142857" title="142857 — chinês" lang="zh" hreflang="zh" data-title="142857" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q846329#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/142857_(n%C3%BAmero)" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discuss%C3%A3o:142857_(n%C3%BAmero)" rel="discussion" title="Discussão sobre o conteúdo da página [t]" accesskey="t"><span>Discussão</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Mudar a variante da língua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">português</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet 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[h]" accesskey="h"><span>Ver histórico</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramentas de página"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramentas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Ferramentas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Ferramentas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mais opções" > <div class="vector-menu-heading"> Operações </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/142857_(n%C3%BAmero)"><span>Ler</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=142857_(n%C3%BAmero)&veaction=edit" title="Editar esta página [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=142857_(n%C3%BAmero)&action=edit" title="Editar o código-fonte desta página [e]" accesskey="e"><span>Editar código-fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=142857_(n%C3%BAmero)&action=history"><span>Ver histórico</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Geral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_afluentes/142857_(n%C3%BAmero)" title="Lista de todas as páginas que contêm hiperligações para esta [j]" accesskey="j"><span>Páginas afluentes</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Altera%C3%A7%C3%B5es_relacionadas/142857_(n%C3%BAmero)" rel="nofollow" title="Mudanças recentes nas páginas para as quais esta contém hiperligações [k]" accesskey="k"><span>Alterações relacionadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:Carregar_ficheiro" title="Carregar ficheiros [u]" accesskey="u"><span>Carregar ficheiro</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginas_especiais" title="Lista de páginas especiais [q]" accesskey="q"><span>Páginas especiais</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=142857_(n%C3%BAmero)&oldid=52953338" title="Hiperligação permanente para esta revisão desta 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style="width: 20em; text-align: left; font-size: 88%;"> <tbody><tr> <th colspan="2" class="topo padrao" style="text-align:center; background-color:#B0C4DE"><span class=""><span style="font-size:132%;">142857</span></span> </th></tr> <tr> <th colspan="2" style="text-align:center; background:LightSteelBlue;"><a href="/wiki/Lista_dos_numerais" class="mw-redirect" title="Lista dos numerais">Nomes dos numerais</a> </th></tr> <tr> <td scope="row" style="vertical-align: top; text-align: left;"><a href="/wiki/N%C3%BAmero_cardinal" title="Número cardinal">Cardinal</a> </td> <td style="vertical-align: top; text-align: left;"><a href="https://pt.wiktionary.org/wiki/142857_(n%C3%BAmero)" class="extiw" title="wikt:142857 (número)">142857 (número)</a> </td></tr> <tr> <td scope="row" style="vertical-align: top; text-align: left;"><a href="/wiki/N%C3%BAmero_ordinal" title="Número ordinal">Ordinal</a> </td> <td style="vertical-align: top; text-align: left;">[[:wikt:Centésimo quadragésimo segundo milésimo<br />octingentésimo quinquagésimo sétimo|Centésimo quadragésimo segundo milésimo<br />octingentésimo quinquagésimo sétimo]] </td></tr> <tr> <th colspan="2" style="text-align:center; background:LightSteelBlue;"><a href="/wiki/Sistema_de_numera%C3%A7%C3%A3o" title="Sistema de numeração">Notações nos principais sistemas</a> </th></tr> <tr> <td scope="row" style="vertical-align: top; text-align: left;"><a href="/wiki/Algarismos_indo-ar%C3%A1bicos" class="mw-redirect" title="Algarismos indo-arábicos">Numeração indo-arábica</a> </td> <td style="vertical-align: top; text-align: left;">142857 </td></tr> <tr> <td scope="row" style="vertical-align: top; text-align: left;"><a href="/wiki/Numera%C3%A7%C3%A3o_romana" title="Numeração romana">Numeração romana</a> </td> <td style="vertical-align: top; text-align: left;"><span style="text-decoration:overline;">CXLII</span>DCCCLVII </td></tr> <tr> <td scope="row" style="vertical-align: top; text-align: left;"><a href="/wiki/Sistema_bin%C3%A1rio_(matem%C3%A1tica)" class="mw-redirect" title="Sistema binário (matemática)">Sistema binário</a> </td> <td style="vertical-align: top; text-align: left;">100010111000001001 </td></tr> <tr> <td scope="row" style="vertical-align: top; text-align: left;"><a href="/wiki/Sistema_hexadecimal" class="mw-redirect" title="Sistema hexadecimal">Sistema hexadecimal</a> </td> <td style="vertical-align: top; text-align: left;">22E09 </td></tr> <tr> <th colspan="2" style="text-align:center; background:LightSteelBlue;">Propriedades matemáticas </th></tr> <tr> <td scope="row" style="vertical-align: top; text-align: left;"><a href="/wiki/Fatoriza%C3%A7%C3%A3o" class="mw-redirect" title="Fatorização">Fatorização</a> </td> <td style="vertical-align: top; text-align: left;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 142857=3^{3}\cdot 11\cdot 13\cdot 37}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>142857</mn> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>11</mn> <mo>⋅</mo> <mn>13</mn> <mo>⋅</mo> <mn>37</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 142857=3^{3}\cdot 11\cdot 13\cdot 37}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3c95bb93435455cb128b6dd592601cc0c4fa8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.302ex; height:2.676ex;" alt="{\displaystyle 142857=3^{3}\cdot 11\cdot 13\cdot 37}"></span> </td></tr> <tr> <td style="vertical-align: top; text-align: center;" colspan="2"> <table style="width:100%;"> <tbody><tr> <td> </td> <td> </td> <td> </td></tr> <tr> <td> </td> <td> </td> <td> </td></tr></tbody></table> </td></tr> <tr> <th colspan="2" style="text-align:center; background:LightSteelBlue;">Lista de números <a href="/wiki/N%C3%BAmero_inteiro" title="Número inteiro">inteiros</a> </th></tr> <tr> <td style="vertical-align: top; text-align: center;" colspan="2"><a href="/wiki/Cem_mil" class="mw-redirect" title="Cem mil">100000</a> <a href="/wiki/Cento_e_quarenta_e_dois_mil_oitocentos_e_cinquenta_e_sete" class="mw-redirect" title="Cento e quarenta e dois mil oitocentos e cinquenta e sete">142857</a> </td></tr> </tbody></table> <p>142.857 é um <a href="/wiki/N%C3%BAmero_c%C3%ADclico" title="Número cíclico">número cíclico</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Permutações_circulares"><span id="Permuta.C3.A7.C3.B5es_circulares"></span>Permutações circulares</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=142857_(n%C3%BAmero)&veaction=edit&section=1" title="Editar secção: Permutações circulares" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=142857_(n%C3%BAmero)&action=edit&section=1" title="Editar código-fonte da secção: Permutações circulares"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ao multiplicá-lo por 2, 3, 4, 5, ou 6 você terá uma permutação cíclica do número, como a seguir:<sup id="cite_ref-leybourn.p.137_1-0" class="reference"><a href="#cite_note-leybourn.p.137-1"><span>[</span>1<span>]</span></a></sup> </p> <dl><dd>1 x 142 857 = <b>1</b>42 857</dd> <dd>2 x 142 857 = 285 7<b>1</b>4</dd> <dd>3 x 142 857 = 428 57<b>1</b></dd> <dd>4 x 142 857 = 57<b>1</b> 428</dd> <dd>5 x 142 857 = 7<b>1</b>4 285</dd> <dd>6 x 142 857 = 857 <b>1</b>42</dd> <dd>7 × 142 857 = 999.999</dd></dl> <p>Ao continuar multiplicando, os resultados serão alguma coisa parecida com 142.857 ou 999.999. Assim: </p> <dl><dd><ul><li>8 × 142 857 = 1.142.856 (Adicione o primeiro dígito ao último e os últimos seis dígitos serão 142 857.)</li> <li>42 × 142 857 = 5.999.994 (Adicione o primeiro dígito ao último, e os últimos seis dígitos serão 999.999.)</li> <li>142 857 × 142 857 = 20.408.122.449 (Adicione os primeiros cinco dígitos aos seis últimos e os últimos seis dígitos serão 142.857.)</li></ul></dd></dl> <p>Ao dividir por 2 ou por 5, você obtém uma permutação cíclica com uma casa decimal: </p> <dl><dd>142.857 ÷ 2 = 71.428,5</dd> <dd>142.857 ÷ 5 = 28.571,4</dd></dl> <p>Ao elevar ao quadrado os três últimos dígitos e subtrair do quadrado dos três primeiros dígitos, você também obtém uma permutação cíclica do número. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 857^{2}=734449\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>857</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>734449</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 857^{2}=734449\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1da61564d2a190d2c190b49304a7e1a4bb008b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.002ex; height:2.676ex;" alt="{\displaystyle 857^{2}=734449\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 142^{2}=20164\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>142</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>20164</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 142^{2}=20164\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11bce9646797dd54e4772a924face2f0bac16a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.839ex; height:2.676ex;" alt="{\displaystyle 142^{2}=20164\,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 734449-20164=714285\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>734449</mn> <mo>−</mo> <mn>20164</mn> <mo>=</mo> <mn>714285</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 734449-20164=714285\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83a4d02d639ab90db2ad020c2b280d18ee33b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.088ex; height:2.343ex;" alt="{\displaystyle 734449-20164=714285\,}"></span></dd></dl> <p>É a parte que se repete na dízima periódica do número racional 1/7 = 0 . 142857 142857 14…<sup id="cite_ref-penny.circulating.decimals_2-0" class="reference"><a href="#cite_note-penny.circulating.decimals-2"><span>[</span>2<span>]</span></a></sup> </p><p>Os múltiplos de 1/7 são simplesmente as cópias dos múltiplos correspondentes de 142857: </p> <dl><dd>1/7 = 0 . 142857 142857 142857 14…</dd> <dd>2/7 = 0 . 285714 285714 285714 28…</dd> <dd>3/7 = 0 . 428571 428571 428571 42…</dd> <dd>4/7 = 0 . 571428 571428 571428 57…</dd> <dd>5/7 = 0 . 714285 714285 714285 71…</dd> <dd>6/7 = 0 . 857142 857142 857142 85…</dd></dl> <p>326451 é uma sequência de fatores que podem ser utilizados para obter um "giro" dos números, como em uma roleta: </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Roue_de_six.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Roue_de_six.svg/200px-Roue_de_six.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Roue_de_six.svg/300px-Roue_de_six.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Roue_de_six.svg/400px-Roue_de_six.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Círculo representativo da propriedade</figcaption></figure> <dl><dd>142857 x <b>3</b> = 42857<b>1</b></dd> <dd>142857 x <b>2</b> = 28571<b>4</b></dd> <dd>142857 x <b>6</b> = 85714<b>2</b></dd> <dd>142857 x <b>4</b> = 57142<b>8</b></dd> <dd>142857 x <b>5</b> = 71428<b>5</b></dd> <dd>142857 x <b>1</b> = 14285<b>7</b></dd></dl> <p>E aqui, uma "escada": </p> <dl><dd>10 = <b>3</b> + (7 x <b>1</b>)</dd> <dd>100 = <b>2</b> + (7 x <b>14</b>)</dd> <dd>1000 = <b>6</b> + (7 x <b>142</b>)</dd> <dd>10000 = <b>4</b> + (7 x <b>1428</b>)</dd> <dd>100000 = <b>5</b> + (7 x <b>14285</b>)</dd> <dd>1000000 = <b>1</b> + (7 x <b>142857</b>)</dd></dl> <p><br /> </p> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-leybourn.p.137-1"><span class="mw-cite-backlink"><a href="#cite_ref-leybourn.p.137_1-0">↑</a></span> <span class="reference-text"><a href="/w/index.php?title=Thomas_Leybourn&action=edit&redlink=1" class="new" title="Thomas Leybourn (página não existe)">Thomas Leybourn</a>, <i>New series of The mathematical repository, Volume 5</i> (1830), p.137s <a rel="nofollow" class="external text" href="http://books.google.com.br/books?id=HdYLAAAAYAAJ&pg=PA137&f=false">[google books]</a></span> </li> <li id="cite_note-penny.circulating.decimals-2"><span class="mw-cite-backlink"><a href="#cite_ref-penny.circulating.decimals_2-0">↑</a></span> <span class="reference-text"><i><a href="/w/index.php?title=The_Penny_Cyclop%C3%A6dia_of_the_Society_for_the_Diffusion_of_Useful_Knowledge&action=edit&redlink=1" class="new" title="The Penny Cyclopædia of the Society for the Diffusion of Useful Knowledge (página não existe)">The Penny Cyclopædia of the Society for the Diffusion of Useful Knowledge</a></i>, Volumes 7-8 (1837), <i>Circulating decimals</i> <a rel="nofollow" class="external text" href="http://books.google.com.br/books?id=w8dPAAAAMAAJ&pg=PA195&f=false">[google books]</a></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=142857_(número)&oldid=52953338">https://pt.wikipedia.org/w/index.php?title=142857_(número)&oldid=52953338</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:N%C3%BAmeros_inteiros" title="Categoria:Números inteiros">Números inteiros</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_notas_de_rodap%C3%A9_desde_maio_de_2013" title="Categoria:!Artigos que carecem de notas de rodapé desde maio de 2013">!Artigos que carecem de notas de rodapé desde maio de 2013</a></li><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_notas_de_rodap%C3%A9_sem_indica%C3%A7%C3%A3o_de_tema" title="Categoria:!Artigos que carecem de notas de rodapé sem indicação de tema">!Artigos que carecem de notas de rodapé sem indicação de tema</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 04h35min de 21 de agosto de 2018.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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