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# If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?

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If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?

A) 5
B) 5(x-y)
C) 20x
D) 20y
E) 35x

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This question can be solved using elimination:
A: Obviously can be a GCD, if 7x and 4y are co-primes
B: May not be able to divide bot, needs work, more on that later
C: 20x cannot divide 35x, since 35x/20x = 7/4 (not an integer)
D: 20y, obviously divides 20y, can divide 35x if x = 4y; therefore CAN be GCD
E: 35x, obviously divides 35x, can divide 20y if y = 7x; therefore CAN be GCD
Based on information so far: C is a sure shot answer, I would go ahead and mark it
For those interested in finding out why B is incorrect, read on:
lets assume 5(x-y) is GCD of 35x and 20y
therefore; 35x = a*5(x-y) , 20y = b*5(x-y) ; where a and b should NECESSARILY be coprimes
solving both equations:
Equation 1: 7x = a(x-y) => ay = x(a-7) => x/y = a/(a-7)
Equation 2: 4y = b(x-y) => y(4+b) = bx => x/y = (4+b)/b
equating both equations: a/(a-7) = (4+b)/b => ab = (4+b)*(a-7) => 4a-7b-28 = 0
this equaiation has many soutions which are coprimes: (a,b) = (21,8),(35,16),(49,24) and so on.
therefore for these values of a and b, 5(x-y) will definitely be the GCD of 35x and 20y; therefore B is incorrect.
answered Sep 17, 2014 by Student (41 points)
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IMO the answer seems to be C
answered Sep 17, 2014 by Partner (696 points)
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Jyotsna Mehta