{"id":17061,"date":"2016-07-28T21:40:26","date_gmt":"2016-07-28T21:40:26","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=17061"},"modified":"2016-07-28T21:40:26","modified_gmt":"2016-07-28T21:40:26","slug":"number-theory-congruence-another-problem-from-underwood-dudley","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/number-theory-congruence-another-problem-from-underwood-dudley\/","title":{"rendered":"Number theory: congruence: another problem from Underwood Dudley"},"content":{"rendered":"

The tutor investigates a problem involving the remainder of a power.<\/h1>\n

On page 48 of his Elementary Number Theory<\/u>, second edition, Underwood Dudley requests the remainder when 20012001<\/sup> is divided by 26.<\/p>\n

Solution:<\/p>\n

2001 mod 26 = 25 ⇒ 2001 = 26k – 1 for some integer k.<\/p>\n

Therefore,<\/p>\n

20012001<\/sup> = (26k-1)2001<\/sup>=(26k-1)(26k-1)(26k-1)…(26k-1)<\/p>\n

where there are 2001 brackets of (26k-1) forming the product.<\/p>\n

Expanding that product, every term is of the form (26k)m<\/sup>(-1)2001-m<\/sup>. Of those, the only term not divisible by 26 will be (26k)0<\/sup>(-1)2001<\/sup> = -1. Therefore,<\/p>\n

20012001<\/sup> = 26q – 1 for some integer q.<\/p>\n

⇒20012001<\/sup> mod 26 = -1.<\/p>\n

Now, -1 ≡ 25 mod 26 (because adding 26 to -1, you get 25).<\/p>\n

So, 20012001<\/sup> mod 26 = 25. Indeed, the remainder when 20012001<\/sup> is divided by 26 is 25.<\/p>\n

Source:<\/p>\n

Dudley, Underwood. Elementary Number Theory,<\/u> second ed. New York: W H Freeman and Company, 1978.<\/p>\n

Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"

The tutor investigates a problem involving the remainder of a power. On page 48 of his Elementary Number Theory, second edition, Underwood Dudley requests the remainder when 20012001 is divided by 26. Solution: 2001 mod 26 = 25 ⇒ 2001 …<\/p>\n

Number theory: congruence: another problem from Underwood Dudley<\/span> Read More »<\/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":[1755,1754,1751],"class_list":["post-17061","post","type-post","status-publish","format-standard","hentry","category-math","category-number-theory","tag-congruence-mod-26","tag-page-48-number-8-elementary-number-theory","tag-underwood-dudley"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17061","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=17061"}],"version-history":[{"count":10,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17061\/revisions"}],"predecessor-version":[{"id":17071,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/17061\/revisions\/17071"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=17061"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=17061"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=17061"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}