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Unformatted text preview: MAT 115A University of California Fall 2010 Solutions Homework 3 1. Rosen 3.3 #20, pg. 95 Let a 1 ,a 2 ,...,a n be integers not all equal to zero. Is it true that the greatest common divisor of these integers ( a 1 ,...,a n ) is the least posi- tive integer of the form m 1 a 1 + m 2 a 2 + + m n a n where m 1 ,...,m n Z ? If so, prove it. If not, give a counterexample. Solution: The statement is true. Let d be the smallest positive linear combination of the a k . Let us write (1) d = m 1 a 1 + + m n a n with m i Z . We claim that d | a i for all 1 i n . To prove this use the division algorithm to write a i = q i d + r i where 0 r i &lt; d . Then r i = a i- q i d = (1- m i ) a i- n X k =1 ,k 6 = i m k a k is a linear combination of the a k . Since 0 r i &lt; d and d was the smallest positive such linear combination, we conclude that r i = 0. Hence d | a i for all 1 i n ....
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