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# assignment solutions - MATH 135 Assignment 1 This...

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MATH 135 Assignment 1 This assignment is due at 8:30am on Wednesday January 13, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. Let a, b, c, d be integers. Suppose that c | a and d | b . Prove that cd | ab . Solution: Since c | a there exists an integer q such that a = cq . Since d | b there exists an integer r such that b = dr . Therefore ab = cd ( rq ), so cd | ab by deﬁnition (because qr is an integer). 2. Let a and b be integers. Prove that gcd ( a, b ) | gcd ( a + b, a - b ) . Solution: Let d = gcd ( a + b, a - b ). Then we know there exist integers x and y such that ( a + b ) x + ( a - b ) y = d. Therefore a ( x + y ) + b ( x - y ) = d. By deﬁnition gcd ( a, b ) | a and gcd ( a, b ) | b . Therefore gcd ( a, b ) | d , as required. 3. Let a, b, c be integers. Suppose that gcd ( a, b ) = 1 and c | ( a + b ). Prove that gcd ( a, c ) = 1. Solution: Let d = gcd ( a, c ). Then by deﬁnition d | a and d | c , and d 0. Since

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assignment solutions - MATH 135 Assignment 1 This...

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