algebric approach?

[ildude02]

I was wondering if there is an effective algebric approach without having to pick numbers to solve this question, especially while considering both the statements together.

if x and y are positive integers, is 2x a multiple of y?

1) 2x + 2 is a multiple of y
2) y is a multiple of x

Appreciate anyone's response.

[Ian Stewart]

I was wondering if there is an effective algebric approach without having to pick numbers to solve this question, especially while considering both the statements together.

if x and y are positive integers, is 2x a multiple of y?

1) 2x + 2 is a multiple of y
2) y is a multiple of x

Appreciate anyone's response.

When you look at both together, you know that:

2x+2 is a multiple of y, and y is a multiple of x.

So 2x+2 is a multiple of x. Algebraically, this means there is an integer k for which

2x + 2 = kx
2 = kx - 2x
2 = (k-2)x

Since k-2 and x must both be positive integers, and since their product is 2, one of them must be equal to 1, and the other must be equal to 2. So we have two possibilities for x:

x=2. Then 2x+2 = 6. If y is a multiple of 2 and a divisor of 6, then y = 2 or 6. Is 2x = 4 a multiple of y? No if y=6, yes if y=2. So already we know that the answer is E. If instead you consider the second possibility, you also find the answer is E:

x = 1. Then 2x+2 = 4, and y could be any divisor of 4; y could equal 1, 2 or 4. Again, since y might be 4, 2x might not be a multiple of y, so the statements together are not sufficient.

[ildude02]

Thank you!