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Unformatted text preview: 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....
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This homework help was uploaded on 04/02/2008 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at University of California, Berkeley.
 Fall '07
 COURTNEY
 Math

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