{"id":17885,"date":"2016-09-19T00:54:42","date_gmt":"2016-09-19T00:54:42","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=17885"},"modified":"2016-09-19T00:54:42","modified_gmt":"2016-09-19T00:54:42","slug":"number-theory-fermats-theorem","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/number-theory-fermats-theorem\/","title":{"rendered":"Number theory:  Fermat&#8217;s Theorem"},"content":{"rendered":"<h1>The tutor introduces Fermat&#8217;s Theorem with a first example.<\/h1>\n<p>Fermat&#8217;s Theorem states that, for a prime number p and a number b not a multiple of p,<\/p>\n<p style=\"text-align:center\">b<sup>p-1<\/sup> &#8801; 1 (mod p).<\/p>\n<p>(See my post <a href=\"?p=3892\">here<\/a> for a working definition of mod.)<\/p>\n<p>This theorem is very useful for solving the following type of question:<\/p>\n<p><strong>Example:  Give the remainder when 16^70 is divided by 71.<\/strong><\/p>\n<p>Solution:  71 is prime, and 16 is clearly not a multiple of it.  Therefore, by Fermat&#8217;s Theorem,<\/p>\n<p>16^70 &#8801; 1 (mod 71)<\/p>\n<p>Therefore, the remainder when 16^70 is divided by 71 is 1.<\/p>\n<p>Source:<\/p>\n<p>Dudley, Underwood.  <u>Elementary Number Theory<\/u>.  New York:  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 introduces Fermat&#8217;s Theorem with a first example. Fermat&#8217;s Theorem states that, for a prime number p and a number b not a multiple of p, bp-1 &#8801; 1 (mod p). (See my post here for a working definition &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/number-theory-fermats-theorem\/\"> <span class=\"screen-reader-text\">Number theory:  Fermat&#8217;s Theorem<\/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":[1896],"class_list":["post-17885","post","type-post","status-publish","format-standard","hentry","category-math","category-number-theory","tag-fermats-theorem-ap-1--1-mod-p"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17885","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=17885"}],"version-history":[{"count":6,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17885\/revisions"}],"predecessor-version":[{"id":17891,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17885\/revisions\/17891"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=17885"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=17885"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=17885"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}