WA 6 - Nicola Hodges David Walnut MATH 290-002 October 23rd...

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Nicola Hodges David Walnut MATH 290-002 October 23 rd , 2010 Writing Assignment 6 1. Let a and b be natural numbers, and let d be the smallest natural number such that there exists integers x and y such that ax+by = d. Prove that d = GCD (a,b). (Hint: Letting c=GCD (a,b), first prove that c d by showing that c/d. Then show that d c by showing that d is a common divisor of a and b) Let a and b be natural numbers and d be the smallest natural number such that there exists integers x and y such that ax +by = d. We will show that d = GCD (a,b). Let c=GCD(a,b), so c divides a and c divides b and there is no other greater divisor. We will show c and d are equivalent. First show, c d. Since c divides a, there exists an integer l such that cl=a. Since c divides b, there exists an integer k such that ck=b Note, we assumed ax + by = d. Through substitution, ax + by = (cl)x + (ck)y = d So, c(lx + ky) = d. Since, lx+ky is an integer, c divides d. By definition, since c divides d, then c d. Next show, d c. Let a and b be natural numbers. Consider the set S of all positive integers of the form am+bn, where m and n are integers. S is nonempty and by the well ordering principle, there is a least member, d=ax+by.
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