{"id":16490,"date":"2016-07-02T15:15:56","date_gmt":"2016-07-02T15:15:56","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=16490"},"modified":"2016-07-04T15:10:42","modified_gmt":"2016-07-04T15:10:42","slug":"math-number-theory-n-mod-3%c2%b2","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/math-number-theory-n-mod-3%c2%b2\/","title":{"rendered":"Math:  number theory:  (n mod 3)\u00b2"},"content":{"rendered":"<h1>The tutor shows an interesting consequence of mod 3 arithmetic.<\/h1>\n<p>Back in my <a href=\"?p=3892\">March 25, 2014 post,<\/a> I mentioned that <em>mod<\/em> means <em>remainder<\/em>.  For example, 19 mod 4 = 3, because when you divide 19 by 4, you get 3 left over.<\/p>\n<p><u>Claim<\/u>:  If neither of two numbers is divisible by 3, the difference of their squares must be.<\/p>\n<p><u>Proof<\/u>:<\/p>\n<p>If a number n is not divisible by 3, then either n mod 3 = 2 or n mod 3 = 1.<\/p>\n<p>If n mod 3 = 2, then n = 3x+2 for some integer x.  n\u00b2, then, is<\/p>\n<p style=\"text-align:center\"> (3x+2)\u00b2=(3x+2)(3x+2)=9x\u00b2+6x+4.  9x\u00b2+6x+4=9x\u00b2+6x+3+1=3(3x\u00b2+2x+1) + 1.<\/p>\n<p>If, on the other hand, n=3y+1, then n\u00b2=(3y+1)\u00b2=(3y+1)(3y+1)=9y\u00b2+6y+1=3(3y\u00b2+2y)+1.<\/p>\n<p>Therefore, if two numbers m and n are both indivisible by 3, then m\u00b2-n\u00b2 has the form 3p+1 &#8211; (3q+1) = 3p+1-3q-1=3p-3q=3(p-q).  The difference of the squares of m and n must be divisible by 3.<\/p>\n<p>Source:<\/p>\n<p>Dudley, Underwood.  <u>Elementary Number Theory<\/u>. New York:<br \/>&nbsp;&nbsp;W H Freeman and Company, 1978.<\/p>\n<p>Jack of <a href=\"https:\/\/www.oracletutoring.ca\">Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The tutor shows an interesting consequence of mod 3 arithmetic. Back in my March 25, 2014 post, I mentioned that mod means remainder. For example, 19 mod 4 = 3, because when you divide 19 by 4, you get 3 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/math-number-theory-n-mod-3%c2%b2\/\"> <span class=\"screen-reader-text\">Math:  number theory:  (n mod 3)\u00b2<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,1677],"tags":[1687,1688,1686],"class_list":["post-16490","post","type-post","status-publish","format-standard","hentry","category-math","category-number-theory","tag-congruence-mod-3","tag-difference-of-squares-mod-3","tag-mod-arithmetic"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16490","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=16490"}],"version-history":[{"count":21,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16490\/revisions"}],"predecessor-version":[{"id":16529,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/16490\/revisions\/16529"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=16490"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=16490"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=16490"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}