{"id":377,"date":"2020-10-02T14:33:08","date_gmt":"2020-10-02T19:33:08","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=377"},"modified":"2020-10-16T13:49:51","modified_gmt":"2020-10-16T18:49:51","slug":"problem-3-fall-2020-divisible-by-9","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2020\/10\/02\/problem-3-fall-2020-divisible-by-9\/","title":{"rendered":"Problem 3 &#8211; Fall 2020 &#8211; Divisible by 9"},"content":{"rendered":"<p><span style=\"font-size: xx-large\"><b>Divisible by 9<\/b><\/span><\/p>\n<p><span style=\"font-size: small\"><i>This problem is from Alexander Karabegov.<\/i><\/span><\/p>\n<p>Given an integer <i>m<\/i>, suppose that the integer <i>n<\/i> is obtained from <i>m<\/i> by permuting its digits. For example, if <i>m<\/i> = 171243, then <i>n<\/i> could be 711234, or <i>n<\/i> could be 432117, and so on. Show that the difference <i>m<\/i> &#8211; <i>n<\/i> is divisible by 9.<\/p>\n<p>Please submit all response to mathpotw@acu.edu by 5:00 PM on Thursday.<\/p>\n<p><span style=\"font-size: small\"><i>Correct solutions were submitted by:\u00a0 none.<\/i><\/span><\/p>\n<p>The outline of the solution we present is to show that <i>m<\/i> is congruent to the sum of its digits modulo 9, and that <i>n<\/i> is congruent to the sum of its digits modulo 9. Then since the digits of <i>n<\/i> are just a rearrangement of the digits of <i>m<\/i>, both sums of the digits are the same. We then apply the transitive property of congruence: if <i>x<\/i> is congruent to <i>y<\/i> and <i>y<\/i> is congruent to <i>z<\/i>, then <i>x<\/i> is congruent to <i>z<\/i>.<\/p>\n<p>Recall that two integers are congruent modulo 9 if and only if the difference of the two numbers is divisible by 9. So we look at the difference between an integer and the sum of its digits.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/math.acu.edu\/green\/pow\/20064\/sol3eq2.gif\" width=\"192\" height=\"55\" \/>Now it is clear that 9 divides this difference. The problem is solved.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Divisible by 9 This problem is from Alexander Karabegov. Given an integer m, suppose that the integer n is obtained from m by permuting its digits. For example, if m = 171243, then n could be 711234, or n could be 432117, and so on. Show that the difference m &#8211; n is divisible by [&hellip;]<\/p>\n","protected":false},"author":130,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[181671],"tags":[],"class_list":["post-377","post","type-post","status-publish","format-standard","hentry","category-mu-sigma-events"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/377","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=377"}],"version-history":[{"count":3,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/377\/revisions"}],"predecessor-version":[{"id":392,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/377\/revisions\/392"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=377"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=377"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}