Problem Solving

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

a. 5
b.5(x-y)
c.20x
d. 20y
e.35x

Solution that is given to me is not making sense. Is there a way to pick a number for this situation and test the given answer choices. MY instincts say to say 700xy. Am I on right track? Is there an easier way to do this with picking numbers?



qa is C

Enginpasa1 wrote:If x and y are positive integers, which of the following CANNOT be the greatest common divisior of 35x and 20y?

a. 5
b.5(x-y)
c.20x
d. 20y
e.35x

Solution that is given to me is not making sense. Is there a way to pick a number for this situation and test the given answer choices. MY instincts say to say 700xy. Am I on right track? Is there an easier way to do this with picking numbers?



qa is C

I don't know how to use numbers for this question, but you can apply logic to get to the right answer.

Using properties of GCD (every common divisor of a and b is a divisor of gcd(a,b), we can cross out choices A, D and E.

For choice B we know that anything times 5 will be a divisor of 35x and 20y since 5 is one of the prime factor of 35 and 20, leaving us with Choice C as the answer.

Hope I made sense...

Enginpasa1 wrote:If x and y are positive integers, which of the following CANNOT be the greatest common divisior of 35x and 20y?

a. 5
b.5(x-y)
c.20x
d. 20y
e.35x

Solution that is given to me is not making sense. Is there a way to pick a number for this situation and test the given answer choices. MY instincts say to say 700xy. Am I on right track? Is there an easier way to do this with picking numbers?

qa is C

I think one problem you're having is how you interpreted the question stem.

We're looking for a number than CANNOT be the greatest common divisor of 35x and 20y. In other words, which choice CANNOT be the biggest number that goes into BOTH 35x and 20y. We look at each of those two terms separately - we're not supposed to add, multiply, subtract or divide them.

The 4 wrong choices all COULD be the GCD of both terms. In other words, for 4 of the choices:

35x/choice and 20y/choice will be an integer.

(a) 5: goes into both 35 and 20, so definitely a divisor of both. Could it be the GCD of both? Sure, let x=y=1 and 5 is the GCD of 35 and 20.

(b) 5(x-y): does it have to go into both? No, but can we make it go into both? Sure: if we pick x=2 and y=1, then we're left with 5(1) = 5, which is the GCD of 70 and 20.

[note: preciousrain7's explanation of (b) is incorrect, since it isn't true that ANYTHING times 5 will be a divisor of 35x and 20y... for example, 10000*5 doesn't necessarily go into either one.]

(c) 20x: 20x CANNOT possibly be a factor of 35x, since if we write it as a fraction we get:

35x/20x = 35/20 = 7/4 which isn't an integer.

Therefore, 20x CANNOT possibly be the GCD of 35x and 20y.

thanks for the post Stuart.

one small error in your explanation in part b though, the Greatest Common Denominator of 70 and 20 is 10 and not 5, isn't it?

Other than that, it was flawless. Great job! I had no idea how to do this on my own.