Homework 11

Homework 11 - )) ( D ( a ) D ( b )) . 2(e). Prove or...

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MATH 74 HOMEWORK 11 (DUE FRIDAY DECEMBER 7) Note the nonstandard due date. 1. Suppose that a 1 ,a 2 ,b 1 ,b 2 are positive integers, that (1) gcd( a 1 ,b 1 ) = 1 , that (2) gcd( a 2 ,b 2 ) = 1 , and that (3) a 1 b 2 = a 2 b 1 . Prove that a 1 = a 2 and that b 1 = b 2 . 2(a). Compute, without proof, the set D (216). 2(b). Compute, again without proof, the set D (90), and then use that to compute D (216) D (90). 2(c). Assuming you have done 2(b) correctly, you will note that D (216) D (90) is D ( n ) for a certain positive integer n . What is n ? 2(d). Prove or disprove: for any positive integers a and b , one has D (gcd( a,b
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Unformatted text preview: )) ( D ( a ) D ( b )) . 2(e). Prove or disprove: for any positive integers a and b , one has ( D ( a ) D ( b )) D (gcd( a,b )) . 3. Find integers M and N with the property that 213 M + 14 N = 1 , or explain why there are no such integers. 4(a). Give an example of integers a,b,c with the property that a | c and b | c but ab 6 | c . 4(b). Prove that if a,b,c are integers satisfying a | c and b | c and gcd( a,b ) = 1, then ab | c ....
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