The other week, a colleague walked into my office and posed the following problem to me:<\/p>\n
For all n where n is prime and n > 3, show that n² – 1 is divisible by 24.
\nExamples: 5² – 1 = 24; 7² -1 = 48; 11² – 1 = 120.<\/p>\n
(If you’re interested, give it a go before looking at the solution. It doesn’t require any advanced Maths.)<\/p>\n
First, what does it mean to be divisible by 24? Well, it means being divisible by 2, 2, 2, and 3 (the prime factorization of 24). So we have to find a factor of 3, and 3 factors of 2.<\/p>\n
As any highschool mathematician knows, n² – 1 = (n + 1)(n – 1).<\/p>\n
Since n is prime and n > 3, it must be odd. Which means that both (n + 1) and (n – 1) must be even (divisible by 2). Moreover, since they are consecutive multiples of 2, one of them must be divisible by 4, and the other by 2.<\/p>\n
We also know that (n – 1), n, (n + 1) are three consecutive numbers, so one of them must be divisible by 3. Since n is prime and n > 3, it cannot be divisible by 3. So either (n – 1) or (n + 1) must be divisible by 3.<\/p>\n
Which means that (n + 1)(n – 1) is divisible by 24. QED.<\/p>\n","protected":false},"excerpt":{"rendered":"
The other week, a colleague walked into my office and posed the following problem to me: For all n where n is prime and n > 3, show that n² – 1 is divisible by 24. Examples: 5² – 1 = 24; 7² -1 = 48; 11² – 1 =…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-794","post","type-post","status-publish","format-standard","hentry","category-maths"],"_links":{"self":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/794","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=794"}],"version-history":[{"count":1,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/794\/revisions"}],"predecessor-version":[{"id":795,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/794\/revisions\/795"}],"wp:attachment":[{"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=794"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=794"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.elbeno.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}