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74-hw3 - MATH 74 HOMEWORK 3 DUE MONDAY 2/12 1 If a and b...

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MATH 74 HOMEWORK 3: DUE MONDAY 2/12 1. If a and b are natural numbers, and b > a , and a | b , then there are q N and 0 < r < a satisfying b = qa + r. In class we proved gcd( a, b ) = gcd( a, r ) by iterating a subtration rule gcd( a, b ) = gcd( a, b - a ) repeatedly. In this exercise you will give a different, more direct proof. 1(a). Let a, b, q, r be as above. Suppose you have a natural number x that is in D ( a, b ). Show, making explicit references to the definitions of the relevant things, that x is also an element of D ( a, r ). 1(b). Give a similarly explicit proof that if x is in D ( a, r ) then x is in D ( a, b ). Remark. The D ( , ) notation is explained in the lecture outlines, if you missed it. Together, the statements 1(a) and 1(b) show that D ( a, r ) = D ( a, b ) , and then gcd( a, r ) = gcd( a, b ) follows from the definition of gcd . Remark. In the previous exercise, you showed that two pairs of numbers had the same gcd by showing that they had the same set of common divisors. This raises
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