h2 - for if c = p 2 e 1 1 · · · p 2 e n n , then u = p e...

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1 Homework II: June 23, 2009 1 . 58 . Prove that if gcd( r, m ) = 1 = gcd( r 0 , m ), then gcd( rr 0 , m ) = 1. Since gcd( r, m ) = 1, there are integers s, t with 1 = sr + tm ; since gcd( r 0 , m ) = 1, there are integers s 0 , t 0 with 1 = s 0 r 0 + t 0 m . Multiplying, 1 = ( sr + tm )( s 0 r 0 + t 0 m ) = ( ss 0 ) rr 0 + [ srt 0 + ts 0 r 0 + tt 0 m ) m. Therefore, 1 is a linear combination of rr 0 and m . As 1 is, obviously, the smallest positive linear combination, it must be the gcd. 1 . 71 . If a and b are positive integers with gcd( a, b ) = 1 and ab is a square, prove that both a and b are squares. Note that if a positive integer c is a square; that is, c = d 2 , then all the exponents in its prime factorization are even: if d = p e 1 1 · · · p e n n , then c = d 2 = p 2 e 1 1 · · · p 2 e n n . Conversely, if all the exponents of c are even, then c is a square,
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Unformatted text preview: for if c = p 2 e 1 1 · · · p 2 e n n , then u = p e 1 1 · · · p e n n is an integer, and u 2 = c . Now consider the prime factorizations a = p e 1 1 · · · p e n n and b = q f 1 1 · · · q f m m . Since a and b are relatively prime, the sets of prime divisors occurring in their prime factorizations are disjoint; that is, the p i 6 = q j for all i, j . Hence, the prime factorization of ab is ab = p e 1 1 · · · p e n n q f 1 1 · · · q f m m . If ab is a square, then all the exponents e i , f j must be even; that is, a and b are squares....
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This note was uploaded on 11/15/2010 for the course MATH 417 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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