Unformatted text preview: b modulo n ” and we write a ³ b . mod n/ if a ± b is divisible by n . The relation a ³ b . mod n/ has the following properties: Lemma 1.7.2. (a) For all a 2 Z , a ³ a . mod n/ . (b) For all a;b 2 Z , a ³ b . mod n/ if, and only if, b ³ a . mod n/ . (c) For all a;b;c 2 Z , if a ³ b . mod n/ and b ³ c . mod n/ , then a ³ c . mod n/ . Proof. For (a), a ± a D is divisible by n . For (b), a ± b is divisible by n if, and only if, b ± a is divisible by n . Finally, if a ± b and b ± c are both divisible by n , then also a ± c D .a ± b/ C .b ± c/ is divisible by n . n For each integer a , write ŒaŁ D f b 2 Z W a ³ b . mod n/ g D f a C kn W k 2 Z g : Note that this is just the set of all integer points that land at the same place as a when the number line is wrapped around the clock face. The set ŒaŁ is called the residue class or congruence class of a modulo n ....
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 Fall '08
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 Algebra, Natural number, Prime number

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