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math 135

# math 135 - 0 Since ab< 0 either a> b< 0 or a< b>...

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Math 135, Winter 2009, Bonus Question Solutions 1 Page 1 Question 1: Show that gcd( ab, c ) =gcd( b, c ) if gcd( a, c ) = 1. Is it true in general that gcd( ab, c ) = gcd( a, c ) · gcd( b, c )? Solution: Let d =gcd( b, c ). Then d | b, d | c implying d | ab and d | c . Hence d is a common divisor of ab and c . Since gcd( a, c ) = 1, there are integers x, y such that ax + cy = 1. Hence abx + bcy = b . Let e | ab and e | c . Then e | ( abx + bcy ) = b . Thus e | b and e | c . Since d =gcd( b, c ), we obtain e d . Therefore e | ab, e | c implies that e d . Since d is a common divisor of ab and c , we get gcd( ab, c ) = d =gcd( b, c ). In general, gcd( ab, c ) =gcd( a, c ) · gcd( b, c ) is not true. Let a = 4 , b = 6 , c = 2. Then gcd( ab, c ) = 2 but gcd( a, c ) = 2 and gcd( b, c ) = 2. Question 2: For what values of a and b , does the equation ax + by = c have an infinite number of positive solutions x, y ? In the board, I wrote ’ what values of c ’ in place of ’ what values of a and b ’. Solution: Let c be given. Let a, b Z be such that d =gcd( a, b ) | c and ab < 0. Then ax + by = c has a solution x = x 0 , y = y 0 and all the other solutions are given by x = x 0 + n b d , y = y 0 - n a d , n Z . Want infinitely n for which
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Unformatted text preview: 0. Since ab < 0, either a > , b < 0 or a < , b > 0. Recall that d > 0. Suppose a < , b > 0. Then-a d > , b d > 0 and hence x = n b d + x > 0 and y = n (-a d ) + y > 0 if n >-x b d and n >-y-a d = y a d . This is true for all n > maximum(-x b d , y a d ) giving inﬁnite number of positive solutions. Suppose a > , b < 0. Then a d > ,-b d > 0 and hence x = (-n )(-b d )+ x > 0 and y = (-n )( a d )+ y > 0 if-n >-x-b d = x b d and-n >-y a d . or if n <-x b d and n <-y a d . This is true for all n < minimum(-x b d ,-y a d ) giving inﬁnite number of positive solutions. Therefore whenever a, b ∈ Z , ab < 0 and gcd( a, b ) | c , we get inﬁnite number of positive solutions. ± Remark: Whenever ab > , we cannot get an inﬁnite number of positive solutions for any c ....
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