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More notes on Congruence - Math 100 Congruences and...

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Math 100 Congruences and Divisibility Martin H. Weissman 1. Congruence as an equivalence relation Recall the definition of “congruence mod n ”: Definition 1. Suppose that a and b are integers, and n is a positive integer. We say “ a is congruent to b , mod n ”, and we write a b (mod n ) if there exists m Z such that a - b = mn . Other interpretations are the following: a is congruent to b , mod n , if the difference a - b is a multiple of n . a is congruent to b , mod n , if the remainder after dividing a by n equals the remainder after dividing b by n . a is congruent to b , mod n , if a is equal to b plus a multiple of n . We begin with the following observation: Theorem 2. Let n be a positive integer. Congruence mod n is an equivalence relation on Z . Proof. Congruence mod n is reflexive, since if a Z , then a - a = 0 · n , and hence a a (mod n ). To prove that congruence mod n is symmetric, suppose that a,b Z , and a b (mod n ). Then a - b = mn , for some m Z . Therefore, b - a = ( - m ) n . Hence b a (mod n ). Hence, congruence mod n is a symmetric relation. To prove that congruence mod n is transitive, suppose that a,b,c Z , and a b (mod n ) and b c (mod n ). Then a - b = mn , for some m Z , and b - c = kn , for some k Z . Therefore, a - c = ( a - b ) + ( b - c ) = ( m + k ) n. Hence a c (mod n ). Therefore, congruence mod n is a transitive relation. / Since congruence mod n is an equivalence relation on Z , it partitions Z into equivalence classes, which are usually called “congruence classes (mod n )”. Proposition 3.
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This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Fall '08 term at UCSC.

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More notes on Congruence - Math 100 Congruences and...

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