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Unformatted text preview: Z * n by a k b for some i N ,k i a b (mod n ) (a) Prove that k is actually an equivalence relation. (b) Let Pow ( k,n ) be the powers of k in Z * n : Pow ( k,n ) = Z * n : for some i N , k i (mod n ) Prove that Pow ( k,n ) is an equivalence class of k . (c) Let X be any equivalence class of k and let a X . Dene f a : Pow ( k,n ) X by f a ( ) = a MOD n . Prove that the range of f a is actually contained in X . Furthermore, prove that f a is a bijection. This tells us that all the equivalence classes of k are the same size. (d) Prove that the size of Pow ( k,n ) divides the size of Z * n . (e) Prove that k  Pow( k,n )  1(mod n ). (f) Conclude that for any k Z * n , k ( n ) 1(mod n )....
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 Summer '08
 STRAIN
 Math

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