MATH 74 HOMEWORK 3: DUE MONDAY 2/121.Ifaandbare natural numbers, andb > a, anda|b, then there areq∈Nand0< r < asatisfyingb=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).Leta, b, q, rbe as above. Suppose you have a natural numberxthat is inD(a, b). Show, making explicit references to the definitions of the relevant things,thatxis also an element ofD(a, r).1(b).Give a similarly explicit proof that ifxis inD(a, r) thenxis inD(a, b).Remark.TheD(,)notation is explained in the lecture outlines, if you missedit. Together, the statements 1(a) and 1(b) show thatD(a, r) =D(a, b), and thengcd(a, r) = gcd(a, b)follows from the definition ofgcd.Remark.In the previous exercise, you showed that two pairs of numbers had thesamegcdby showing that they had the same set of common divisors. This raises
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