Unformatted text preview: Z * n by a âˆ¼ k b â‡â‡’ for some i âˆˆ N ,k i Â· a â‰¡ b (mod n ) (a) Prove that âˆ¼ k is actually an equivalence relation. (b) Let Pow ( k,n ) be the powers of k in Z * n : Pow ( k,n ) = Â³ â€˜ âˆˆ Z * n : for some i âˆˆ N ,â€˜ â‰¡ k i (mod n ) Â´ Prove that Pow ( k,n ) is an equivalence class of âˆ¼ k . (c) Let X be any equivalence class of âˆ¼ k and let a âˆˆ X . Deï¬ne f a : Pow ( k,n ) â†’ X by f a ( â€˜ ) = aâ€˜ 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 âˆ¼ 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 )  â‰¡ 1(mod n ). (f) Conclude that for any k âˆˆ Z * n , k Ï† ( n ) â‰¡ 1(mod n )....
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 Summer '08
 STRAIN
 Math, Equivalence relation, Prime number, positive integer, equivalence class

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