{"id":64,"date":"2005-01-30T20:25:47","date_gmt":"2005-01-30T20:25:47","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=64"},"modified":"2005-01-30T20:25:47","modified_gmt":"2005-01-30T20:25:47","slug":"divisors","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=64","title":{"rendered":"Divisors"},"content":{"rendered":"<p>Most people, when asked how many divisors the number 60 has (including 1 and 60), would struggle to do so without listing them all. Yet once you know that the prime factorisation of 60 is <img src='\/maths\/latexrender\/pictures\/0c2f51a243adb068a9c5c5c0a9c9cac1.gif' title='60=2^2\\times3\\times5' alt='60=2^2\\times3\\times5' align=absmiddle> you can immediately say that the number of divisors is <img src='\/maths\/latexrender\/pictures\/f8956a59d8074f9216caab9fefc80001.gif' title='3 \\times 2 \\times 2=12' alt='3 \\times 2 \\times 2=12' align=absmiddle>. In other words, you take each index, add 1 then multiply them together.<\/p>\n<p>This is easy to see if you list the divisors as <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/69911d5266e94d37891401b92d5261d9.gif' title='\\2^0\\times3^0\\times5^0\\&#10;2^1\\times3^0\\times5^0\\&#10;2^2\\times3^0\\times5^0\\&#10;2^0\\times3^1\\times5^0\\&#10;2^1\\times3^1\\times5^0\\&#10;\\cdots\\&#10;2^2\\times3^1\\times5^1\\' alt='\\2^0\\times3^0\\times5^0\\&#10;2^1\\times3^0\\times5^0\\&#10;2^2\\times3^0\\times5^0\\&#10;2^0\\times3^1\\times5^0\\&#10;2^1\\times3^1\\times5^0\\&#10;\\cdots\\&#10;2^2\\times3^1\\times5^1\\' align=absmiddle><\/ul>\n<p>It&#8217;s then not difficult to prove the general result:<\/p>\n<ul><i>If <img src='\/maths\/latexrender\/pictures\/d7577edf068e7cfba36c1dacacc1027d.gif' title='n=p_1^{\\alpha_1}.p_2^{\\alpha_2}.\\cdots.p_k^{\\alpha_k}' alt='n=p_1^{\\alpha_1}.p_2^{\\alpha_2}.\\cdots.p_k^{\\alpha_k}' align=absmiddle> then the number of divisors of <img src='\/maths\/latexrender\/pictures\/7b8b965ad4bca0e41ab51de7b31363a1.gif' title='n' alt='n' align=absmiddle> is <img src='\/maths\/latexrender\/pictures\/fd34ca86d04bf29abce0722a72db20fd.gif' title='d(n)=\\prod\\limits^k_{i=1}\\left(\\alpha_i + 1}\\right)' alt='d(n)=\\prod\\limits^k_{i=1}\\left(\\alpha_i + 1}\\right)' align=absmiddle><\/i><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Most people, when asked how many divisors the number 60 has (including 1 and 60), would struggle to do so without listing them all. Yet once you know that the prime factorisation of 60 is you can immediately say that the number of divisors is . In other words, you take each index, add 1 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-64","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/64","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=64"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/64\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=64"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=64"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=64"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}