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e1sol - 1 Math 417 Exam I July 2 2009 Solutions 1 If a and...

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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 . Therefore, n = ak = abb , 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: 37 = 5 · 7 + 2 5 = 3 · 2 + 1 . Thus, gcd(7 , 37) = 1 and 1 = 16 · 7 - 3 · 37. 3 . ( ii ) ( 10 points ) Find all solutions to 7 x 3 mod 37.
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