hw5 - Z * n by a k b for some i N ,k i a b (mod n ) (a)...

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Math 55 - Homework 5 Due Friday, July 15, 2011 Directions: You are encouraged to discuss homework with fellow students. However, you should write up all solutions alone . Your answer should be clear enough that it explains to someone who does not already understand the answer why it works. (This is different than just convincing the grader that you understand.) 1. For a positive integer n , define Z * n to be the integers from 1 to n - 1 which are relatively prime to n . That is Z * n = { k Z : 0 k < n and gcd( k,n ) = 1 } . Defined φ ( n ) to be the size of Z * n . For example, Z * 12 = { 1 , 5 , 7 , 11 } and φ (12) = 4. (a) If p is a prime, and i is a positive integer, prove that φ ( p i ) = p i - 1 ( p - 1). (b) If m and are relatively prime, prove that φ ( m‘ ) = φ ( m ) · φ ( ). (c) If n = p e 1 1 p e 2 2 ...p e j j , where the p ’s are distinct primes and the e ’s are positive, prove that φ ( n ) = n ± 1 - 1 p 1 ² ± 1 - 1 p 2 ² ... ± 1 - 1 p j ² . 2. Let n be a positive integer. For any k Z * n , we can define an equivalence relation on
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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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