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Unformatted text preview: Math 55  Homework 5 Solutions Friday, July 15, 2011 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 ( n ) to be the size of Z * n . For example, Z * 12 = { 1 , 5 , 7 , 11 } and (12) = 4. (a) If p is a prime, and i is a positive integer, prove that ( p i ) = p i 1 ( p 1). Since the prime factorization of p i is unique, the only prime which divides p i is p . So the only integers from to p i 1 not relatively prime with p i are multiples of p : ,p, 2 p, 3 p,... ( p i 1 ) p . There are p i 1 of those. So ( p i ) = Z * p i = p i p i 1 = p i 1 ( p 1) . (b) If m and are relatively prime, prove that ( m ) = ( m ) ( ). k Z * m k coprime to m k coprime to m and k MOD m coprime to m and k MOD coprime to ( k MOD m,k MOD ) Z * m Z * By the Chinese Remainder Theorem, the map k ( k MOD m,k MOD ) is a bijection, so ( m ) =  Z * m  =  Z * m   Z *  = ( m ) ( ) (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 ( n ) = n 1 1 p 1 1 1 p 2 ... 1 1 p j ....
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This note was uploaded on 03/14/2012 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.
 Summer '08
 STRAIN
 Integers

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