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4383_Notes_4o1o2

4383_Notes_4o1o2 - Math 4383 Sections 4.1 and 4.2 Page 1 of...

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Math 4383 Sections 4.1 and 4.2 Page 1 of 8 Number Theory Chapter 4 Sections 4.1 and 4.2 – Congruence Carl Friedrich Gauss (1777-1855) Amazing mathematical prodigy Add 1+2+3+4+…+100= His Ph.D. dissertation contained the first full proof of the Fundamental Theorem of Algebra Gauss made many contributions to number theory, including the concept of congruence. Definition: Given an fixed positive integer n and any integers a and b, we say “a is congruent to b modulo n” written ( mod a b n If a-b is a multiple of n. Examples:

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Math 4383 Sections 4.1 and 4.2 Page 2 of 8 Congruence modulo one is trivial – everything is congruent, so we normally assume n>1. Congruence modulo 2 – By the division algorithm, every integer is congruent to one of the integers 0,1,2,…, n-1 modulo n. Integers 0, 1, 2,…, n-1 are called a complete set of residues modulo n Any set of n integers, none of which is congruent to any of the others, is a complete set of residues modulo n.
Math 4383 Sections 4.1 and 4.2 Page 3 of 8 Examples: Theorem 4.1

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