Problem Solving

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

The wording of the question tells us that 4 of the answer choices CAN be the greatest common divisor (GCD) of 35x and 20y, and one of them cannot.

So, might BEGIN by eliminating those answer choices that CAN be the greatest common divisor of 35x and 20y

A) 5
5 is a DIVISOR of both 35x and 20y, but can it be the GCD of 35x and 20y?
YES. If x = 1 and y = 1, then 5 is the GCD of 35x and 20y.
ELIMINATE A

B) 5(x - y)
CAN 5(x - y) be the GCD of 35x and 20y?
YES. If x = 3 and y = 2, then 5(x - y) = 5, 35x = 105 and 20y = 40
Since 5 is the GCD of 105 and 40, we can ELIMINATE B

Aside: It can be tough finding values such that 4 of the answer choices are, indeed, the GCD of 35x and 20y. So, we should also be looking for another approach that shows that an answer choice CANNOT be the GCD of 35x and 20y

C) 20x
CAN 20x be the GCD of 35x and 20y?
NO!
How do we know this?
20x cannot be the greatest common DIVISOR of 35x and 20y, because 20x isn't even a DIVISOR of 35x
Notice that 35x/20x = 35/20 = 7/4
This tells us that 20x cannot be a DIVISOR of 35x, which means 20x cannot be the GCD of 35x and 20y.

Answer: C

Cheers,
Brent