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74-hw3s

# 74-hw3s - MATH 74 HOMEWORK 3 SOLUTIONS 1 We are given that...

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MATH 74 HOMEWORK 3 SOLUTIONS 1. We are given that x D ( a, b ). This means (by definition of D ( a, b )) that x | a and x | b , and this means (by definition of “divides”) that there are k N and l N with a = kx and b = lx . We are supposed to show that x D ( a, r ), ie, that x | a and x | r . As we already know that x | a , it is thus enough to show that x | r . But r = b - qa = lx - qkx = ( l - qk ) x. Note that l - qk is certainly an integer, and it has to be positive since r and x both are, so it is a natural number, and hence x | r . 1(b). If x is in D ( a, r ) then (by definition of D ( a, r )) we have that x | a and x | r . We are supposed to show that x must also be in D ( a, b ), ie, that x | a and x | b . Since we know x | a already, it is thus enough to show that x | b . To do this, note that as x | a there is k N satisfying a = kx , and as x | r there is l N satisfying r = lx . We then see that b = qa + r = qkx + lx = ( qk + l ) x. Since q, k, l are natural numbers, so is qk + l . This verifies that x divides b .

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74-hw3s - MATH 74 HOMEWORK 3 SOLUTIONS 1 We are given that...

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