Factor | Definition, Examples, & Facts

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class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/science/factor-mathematics">factor</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <span class="toc-extra-link selected link-gray-900 mt-15"> Introduction </span> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/science/factor-mathematics/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/factor-mathematics">Related Topics</a> </div> <div class="mb-30 tlr-related-quizzes"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Quizzes </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/define-it-math-terms"> <img loading="lazy" src="https://cdn.britannica.com/01/115001-131-7278E518/Enrico-Fermi-Italian-problem-physics-1950.jpg?w=200&h=200&c=crop" alt="Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/define-it-math-terms" >Define It: Math Terms</a> </div> </div> </div> <div class="mb-30 tlr-read-next"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Read Next </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/unusual-counting-systems"> <img loading="lazy" src="https://cdn.britannica.com/41/191041-131-7ECE668A/bunch-numbers.jpg?w=200&h=200&c=crop" alt="bunch of numbers" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/unusual-counting-systems" >Unusual Counting Systems</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/is-zero-an-even-or-an-odd-number"> <img loading="lazy" src="https://cdn.britannica.com/49/191949-131-3E2AC277/balloon.jpg?w=200&h=200&c=crop" alt="number zero, 0 balloon" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/is-zero-an-even-or-an-odd-number" >Is Zero an Even or an Odd Number?</a> </div> </div> </div> <div class="mb-30 tlr-discover"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Discover </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/new-seven-wonders-of-the-world"> <img loading="lazy" src="https://cdn.britannica.com/36/162636-131-E4AA93A0/Colosseum-Rome-Italy.jpg?w=200&h=200&c=crop" alt="The Colosseum, Rome, Italy. Giant amphitheatre built in Rome under the Flavian emperors. (ancient architecture; architectural ruins)" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/new-seven-wonders-of-the-world" >New Seven Wonders of the World</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/how-many-countries-are-there-in-the-world"> <img loading="lazy" src="https://cdn.britannica.com/27/238527-131-D73B3F08/flagpoles-world-countries.jpg?w=200&h=200&c=crop" alt="Flags of the countries of the world (flagpoles)." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/how-many-countries-are-there-in-the-world" >How Many Countries Are There in the World?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/who-really-invented-baseball"> <img loading="lazy" src="https://cdn.britannica.com/53/212553-131-B8C3D5AF/Baseball-bat.jpg?w=200&h=200&c=crop" alt="An old worn baseball and wood bat" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/who-really-invented-baseball" >Who Really Invented Baseball?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/how-does-wi-fi-work"> <img loading="lazy" src="https://cdn.britannica.com/18/197118-131-00A7D6AF/Computer-Code-Binary-Internet-Password-Data-Technology.jpg?w=200&h=200&c=crop" alt="Binary Computer Code, Binary Code, Internet, Technology, Password, Data" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/how-does-wi-fi-work" >How Does Wi-Fi Work?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/falling-far-from-the-tree-7-brilliant-ways-seeds-and-fruits-are-dispersed"> <img loading="lazy" src="https://cdn.britannica.com/86/158086-131-4A55A23A/Dandelion-seeds-wind-Seed-dispersion.jpg?w=200&h=200&c=crop" alt="Dandelion with seeds blowing in the wind. 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Official portrait Obama" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/10-democrats-who-made-history" >10 Democrats Who Made History</a> </div> </div> </div> </div> </div> </div> <button class="drawerToggle btn position-sticky border btn-xs btn-white btn-circle rounded-sm d-none d-lg-flex " type="button" aria-label="Toggle Drawer"> <em class="material-icons font-18 text-blue" data-icon="keyboard_arrow_left"></em> </button> </div> </div> <div class="col"> <div class="h-100 ml-0 pr-lg-0 "> <div class="h-100 grid gx-0 gx-lg-20"> <div class="h-100 col-sm"> <div class="h-100 infinite-pagination-container d-flex flex-column position-relative"> <div class="position-absolute top-0 h-100 w-100"> <div class="toc-sticky-header d-none d-lg-none bg-gray-50 px-10 px-sm-30 position-sticky w-100 "> <div class="toc-sticky-header-inner-container align-items-center d-flex mx-auto h-100 w-100"> <button class="d-flex d-lg-none btn btn-sm btn-white text-blue border-2 border-gray-100 gtm-mobile-toc-header-button js-sections-button d-lg-none p-10"> <em class="material-icons my-n5 md-icon" data-icon="toc"></em> Contents </button> <div class="header-ai-ask-button-placeholder"></div> <div class="header-ai-summarize-button-placeholder"></div> </div> </div> </div> <div class="grey-box w-100 grey-box-top grey-box-bottom"> <div class="grey-box-content mx-auto w-100"> <script type="application/ld+json"> { "@context" : "https://schema.org", "@type" : "BreadcrumbList", "itemListElement" : [ { "@type" : "ListItem", "position" : 1, "item" : { "@id" : "https://www.britannica.com/browse/Science", "name": "Science" } } , { "@type" : "ListItem", "position" : 2, "item" : { "@id" : "https://www.britannica.com/browse/Mathematics", "name": "Mathematics" } } ] } </script> <nav class="breadcrumb mt-20"> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Science">Science</a> </span> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Mathematics">Mathematics</a> </span> </nav> <div class="page2ref-true topic-content topic-type-REGULAR" data-student-article="false"> <script class="page-description-json" type="application/json"> { "url": "/science/factor-mathematics", "shareUrl": "https://www.britannica.com/science/factor-mathematics", "browserTitle": "Factor | Definition, Examples, & Facts", "firstTopicPage": true, "topicId":200005 } </script> <div class="reading-channel"> <div class="topic-header"> <div class="d-flex align-items-top justify-content-between"> <div class="d-flex flex-column"> <div> <div> <h1>factor</h1></div> </div> <div class="topic-identifier font-16 font-md-20">mathematics</div> </div> </div> <div class="d-none d-sm-flex flex-row"> <div class="mr-10 mb-15"> <button class="ai-ask-button btn border-2 btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400"> Ask the Chatbot a Question </button> </div> <div class="d-block md-topic-tools qa-action-buttons mb-15" data-topic-id="200005"> <button class="js-tooltip btn btn-sm btn-outline-blue border pr-10 border-2 text-nowrap" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="more_vert"></em> More Actions </button> <div class="md-more-popover popover popover-sm p-0 font-14 z-1"> <div> <a class="js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" href="/print/article/200005" target="_blank" > <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="print"></em> Print </a> </div> <div> <button class="js-modal qa-cite-modal-button btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=200005] .md-cite-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="verified"></em> Cite </button> <div class="md-cite-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom mb-15"> <em class="material-icons text-blue mr-10">verified</em>Cite </div> <div class="font-serif"> While every effort has been made to follow citation style rules, there may be some discrepancies. 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Let us know if you have suggestions to improve this article (requires login). </div> <div class="type-menu"> <label for="feedback-type" class="label mb-10">Feedback Type</label> <select id="feedback-type" class="form-select mb-30" name="feedbackTypeId" required> <option value="" selected="selected">Select a type (Required)</option> <option value="1">Factual Correction</option> <option value="2">Spelling/Grammar Correction</option> <option value="3">Link Correction</option> <option value="4">Additional Information</option> <option value="5">Other</option> </select> </div> <label for="feedback" class="label mb-10">Your Feedback</label> <textarea id="feedback" class="form-control mb-30" name="feedback" maxlength="3000" rows="7" required></textarea> <button class="btn btn-blue" type="submit">Submit Feedback</button> </form> <div class="success-messaging d-none mt-30"> <div class="title">Thank you for your feedback</div> <p>Our editors will review what you’ve submitted and determine whether to revise the article.</p> </div> </div> </div> <div class="md-websites-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif pb-15 border-bottom font-weight-bold"> External Websites </div> <div class="pb-20"> <ul class="list-unstyled mt-20 lh-lg"> <li><a class="external" href="https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(OpenStax)/06%3A_Factoring" target="_blank" rel="noopener ">Mathematics LibreTexts - Factoring</a></li> <li><a class="external" href="http://www.mathsisfun.com/definitions/factor.html" target="_blank" rel="noopener ">Maths Is Fun - Factor</a></li> </ul> </div> </div> </div> </div> </div> <div class="toc-header-marker"></div> <button class="ai-ask-button btn border-2 js-header-ai-ask-button d-none btn-sm btn-outline-red-400 border-red-400 mr-0 mr-lg-10 ml-5 ml-sm-10 ml-lg-0 p-10"> Ask the Chatbot a Question </button> <div class="md-byline module-spacing "> <div class="font-serif font-12"> <span class="written-by text-gray-700"> Written and fact-checked by </span> <div class="editor-popover popover p-0"> <a class="d-block p-20 qa-editor-popup gtm-byline font-12 byline-contributor" href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419" > <div class="editor-title font-16 font-weight-bold">The Editors of Encyclopaedia Britannica</div> <div class="editor-description font-12 font-serif mt-5 text-black">Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <a class="byline-edit-history" href="https://www.britannica.com/science/factor-mathematics/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn border-2 ai-ask-button btn border-2 module-spacing btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400 p-10 ml-5"> Ask the Chatbot a Question </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/technology/power-factor" topicid="473330">power factor</a></dd> <dd><a href="/science/divisor" topicid="166741">divisor</a></dd> <dd><a href="/science/multiplication" topicid="397199">multiplication</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/factor-mathematics"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref281856"></span>factor</strong>, in <a href="https://www.britannica.com/science/mathematics" class="md-crosslink " data-show-preview="true">mathematics</a>, a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder. For example, 3 and 6 are factors of 12 because 12 ÷ 3 = 4 exactly and 12 ÷ 6 = 2 exactly. The other factors of 12 are 1, 2, 4, and 12. A positive integer greater than 1, or an algebraic expression, that has only two factors (i.e., itself and 1) is termed <span id="ref190549"></span><a href="https://www.britannica.com/science/prime-number" class="md-crosslink " data-show-preview="true">prime</a>; a positive integer or an algebraic expression that has more than two factors is termed <span id="ref190550"></span><a href="https://www.britannica.com/topic/composite-number" class="md-crosslink ">composite</a>. The prime factors of a number or an algebraic expression are those factors which are prime. By the <span id="ref190551"></span><a href="https://www.britannica.com/science/fundamental-theorem-of-arithmetic" class="md-crosslink " data-show-preview="true">fundamental theorem of arithmetic</a>, except for the order in which the prime factors are written, every whole number larger than 1 can be uniquely expressed as the product of its prime factors; for example, 60 can be written as the product 2·2·3·5.</p><div class="one-good-fact-module"> </div><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">Methods for factoring large whole numbers are of great importance in <a href="https://www.britannica.com/topic/public-key-cryptography" class="md-crosslink autoxref " data-show-preview="true">public-key cryptography</a>, and on such methods rests the security (or lack thereof) of data transmitted over the <a href="https://www.britannica.com/technology/Internet" class="md-crosslink " data-show-preview="true">Internet</a>. Factoring is also a particularly important step in the solution of many algebraic problems. For example, the polynomial equation <em>x</em><sup>2</sup> − <em>x</em> − 2 = 0 can be factored as (<em>x</em> − 2)(<em>x</em> + 1) = 0. Since in an <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="integral" href="https://www.merriam-webster.com/dictionary/integral" data-type="MW">integral</a> domain <em>a</em>·<em>b</em> = 0 implies that either <em>a</em> = 0 or <em>b</em> = 0, the simpler equations <em>x</em> − 2 = 0 and <em>x</em> + 1 = 0 can be solved to yield the two solutions <em>x</em> = 2 and <em>x</em> = −1 of the original equation.</p><span class="marker MOD2 mod-inline"></span></section> <span class="md-signature"><a href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419">The Editors of Encyclopaedia Britannica</a></span> <span class="md-signature">This article was most recently revised and updated by <a href="/editor/Erik-Gregersen/6723">Erik Gregersen</a>.</span><span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article> </div> </div> </div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <script type="text/javascript" id="_informizely_script_tag"> var IzWidget = IzWidget || {}; (function (d) { var scriptElement = d.createElement('script'); scriptElement.type = 'text/javascript'; scriptElement.async = true; scriptElement.src = "https://insitez.blob.core.windows.net/site/f780f33e-a610-4ac2-af81-3eb184037547.js"; var node = d.getElementById('_informizely_script_tag'); node.parentNode.insertBefore(scriptElement, node); } )(document); </script>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":1,"pagesTotal":1,"pageId":200005,"pageLength":267,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-133/js/libs/jquery-3.5.0.min.js?v=3.133.9"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-133/dist/topic-page.js?v=3.133.9' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"B","adLeg":"B","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"MEDIUM","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":true} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

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