e1sol - 1 Math 417 Exam I: July 2, 2009; Solutions 1. If a...

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1 Math 417 Exam I: July 2, 2009; Solutions 1 . If a and b are relatively prime and each of them divides an integer n , prove that their product ab also divides n . Here are two proofs (of course, either one suffices for full credit). By hypothesis, n = ak = b` . Since b | ak , Corollary 1.40 [if ( b, a ) = 1 and b | ak , then b | k ] gives b | k ; that is, k = bb 0 . Therefore, n = ak = abb 0 , and so ab | n . The second proof does not use Corollary 1.40. There are integers s and t with 1 = sa + tb . Hence, n = san + tbn = sab` + tbak = ab ( s` + tk ) . Hence, ab | n . 2 . Let I be a subset of Z such that 1. 0 I ; 2. if a, b I , then a - b I ; 3. if a I and q Z , then qa I . Prove that there is a non-negative d I such that I consists precisely of all the multiples of d . This is Corollary 1.37 in the text. 3 . ( i ) ( 5 points ) Write gcd(7 , 37) as a linear combination of 7 and 37. The Euclidean algorithm ends very quickly here:
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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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e1sol - 1 Math 417 Exam I: July 2, 2009; Solutions 1. If a...

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