CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> greatest common divisor in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> greatest common divisor </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/12066/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="arithmetic">Arithmetic</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+arithmetic+geometry">higher arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/number">number</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/integer+number">integer number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, <a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a>, <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, <a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a>, <a class="existingWikiWord" href="/nlab/show/octonion">octonion</a>, <a class="existingWikiWord" href="/nlab/show/adic+number">adic number</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+number">cardinal number</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal number</a>, <a class="existingWikiWord" href="/nlab/show/surreal+number">surreal number</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/second-order+arithmetic">second-order arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+arithmetic">transfinite arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+arithmetic">cardinal arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+arithmetic">ordinal arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+integer">p-adic integer</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+rational+number">p-adic rational number</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></strong>, <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+scheme">arithmetic scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chow+group">arithmetic Chow group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil-%C3%A9tale+topology+for+arithmetic+schemes">Weil-étale topology for arithmetic schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+cohomology">absolute cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Iwasawa-Tate+theory">Iwasawa-Tate theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adelic+integration">adelic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shtuka">shtuka</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenioid">Frobenioid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Riemann-Roch+theorem">arithmetic Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#remarks'>Remarks</a></li> <ul> <li><a href='#in_the_natural_numbers'>In the natural numbers</a></li> <li><a href='#in_the_integers'>In the integers</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">0, 1, 2, \ldots</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/poset">partially ordered</a> by the <a class="existingWikiWord" href="/nlab/show/divisibility+relation">divisibility relation</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">|</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a|b</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mi>a</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">b = a k</annotation></semantics></math> for some natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. This poset is in fact a <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>. The <em>greatest common divisor</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a, b</annotation></semantics></math> is their <a class="existingWikiWord" href="/nlab/show/meet">meet</a> in this lattice.</p> <h2 id="remarks">Remarks</h2> <h3 id="in_the_natural_numbers">In the natural numbers</h3> <p>Spelled out, this means that the greatest common divisor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">a, b \in \mathbb{N}</annotation></semantics></math>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gcd(a, b)</annotation></semantics></math>, is the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math> uniquely characterized by the following two conditions:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">|</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">d|a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">|</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">d|b</annotation></semantics></math>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">c|a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">c|b</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">c|d</annotation></semantics></math>.</p> </li> </ul> <p>One could equivalently equip the natural numbers with a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi><mo>:</mo><mi>ℕ</mi><mo>×</mo><mi>ℕ</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\gcd:\mathbb{N} \times \mathbb{N} \to \mathbb{N}</annotation></semantics></math> which satisfies the two conditions:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\gcd(a, b)|a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\gcd(a, b)|b</annotation></semantics></math>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">c|a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">c|b</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c|\gcd(a, b)</annotation></semantics></math>.</p> </li> </ul> <p>It is almost but not quite true that “greatest” means greatest with respect to the usual ordering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> is the maximal element with respect to the divisibility ordering, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\gcd(0, 0) = 0</annotation></semantics></math> according to the definition above. However, there is no “greatest” common divisor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> with itself if we construe “greatest” in the sense of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>: every natural number is a common divisor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> with itself, and there is no <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>-greatest natural number!</p> <p>Thus, the convention often seen in textbooks, which replaces the second condition above with</p> <ul> <li>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">c|a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">c|b</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>≤</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">c \leq d</annotation></semantics></math></li> </ul> <p>or</p> <ul> <li>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">c|a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">|</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">c|b</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>≤</mo><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \leq \gcd(a, b)</annotation></semantics></math>.</li> </ul> <p>is slightly more awkward, and certainly less “pure” (mixing two relations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>). It is also less robust, because the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi></mrow><annotation encoding="application/x-tex">gcd</annotation></semantics></math> is at bottom an <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>-theoretic notion: the divisibility order on elements of a <a class="existingWikiWord" href="/nlab/show/principal+ideal+domain">principal ideal domain</a> is a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a> whose posetal collapse is the collection of ideals, ordered oppositely to <a class="existingWikiWord" href="/nlab/show/inclusion">inclusion</a>. Thus, in <a class="existingWikiWord" href="/nlab/show/ring">ring</a>-theoretic contexts where there is no sensible notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>, for example in the ring of <a class="existingWikiWord" href="/nlab/show/Gaussian+integers">Gaussian integers</a>, the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi></mrow><annotation encoding="application/x-tex">gcd</annotation></semantics></math> still makes perfectly good sense if we use the first formulation above, expressed purely in terms of divisibility.</p> <p>From the point of view of principal ideals in a <a class="existingWikiWord" href="/nlab/show/pid">pid</a> or <a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+domain">Bézout domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, the gcd corresponds to taking their <a class="existingWikiWord" href="/nlab/show/join">join</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\gcd(a, b)) = (a) + (b)</annotation></semantics></math>. Thus the <a class="existingWikiWord" href="/nlab/show/Euclidean+algorithm">Euclidean algorithm</a>, which applies generally to <a class="existingWikiWord" href="/nlab/show/Euclidean+domains">Euclidean domains</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, is a way of calculating a generator of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a) + (b)</annotation></semantics></math> which consists of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-linear combinations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>.</p> <h3 id="in_the_integers">In the integers</h3> <p>In the set of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, the greatest common divisor only results in a <a class="existingWikiWord" href="/nlab/show/prelattice">prelattice</a> rather than a lattice, because the divisibility relation is only a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a> rather than a <a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a>, due to the fact that there is more than one element of the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> of the integers.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/least+common+multiple">least common multiple</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hermite+normal+form">Hermite normal form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coprime+integers">coprime integers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GCD+ring">GCD ring</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Greatest_common_divisor">Greatest common divisor</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 27, 2024 at 05:59:55. See the <a href="/nlab/history/greatest+common+divisor" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/greatest+common+divisor" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12066/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/greatest+common+divisor/7" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/greatest+common+divisor" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/greatest+common+divisor" accesskey="S" class="navlink" id="history" rel="nofollow">History (7 revisions)</a> <a href="/nlab/show/greatest+common+divisor/cite" style="color: black">Cite</a> <a href="/nlab/print/greatest+common+divisor" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/greatest+common+divisor" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>