ashiima wrote:
w, x, y, and z are integers. If w > x > y > z > 0, is y a common divisor of w and x?
(1)w/x = z^-1 + x^-1
(2)w^2-wy-2w=0
This is a pretty awkwardly designed question. From Statement 1 we know:
w/x = (1/z) + (1/x)
w = (x/z) + 1
Now we know from the stem that w > x, or, since w and x are integers, that w
> x + 1. Substituting in the expression we just found for w, we have that
(x/z) + 1
> x + 1
x/z
> x
If z is an integer, this could only be true if z = 1 (if z is bigger than 1, clearly the left side of the inequality above will be smaller than the right side). So we know that z=1, and plugging that into our expression for w, we know that
w = (x/z) + 1
w = x + 1
So w is 1 greater than x, and w and x are therefore consecutive integers. The Greatest Common Divisor of any two consecutive positive integers is *always* equal to 1. Since y cannot be equal to 1 (since y > x > 0, and x and y are integers, the smallest possible value of y is 2), y cannot be a common divisor of x and w. So Statement 1 is sufficient.
From Statement 2 we can factor out a w:
w^2-wy-2w=0
w(w - y - 2) = 0
For this product to be 0, one of the factors on the left side must be 0. We know that w is not zero, so w - y - 2 must be zero:
w - y - 2 = 0
w = y + 2
But if w > x > y, and each of these quantities are integers, then if w is exactly 2 greater than y, it must be that w, x and y are three consecutive integers. So again we know that w and x are consecutive integers, and just as we saw in Statement 1, it is impossible for y to be a common divisor of both.
So the answer is D, since each statement is sufficient to give a 'no' answer to the question.
I find it an awkward question because it tries to confuse the fundamental concept behind the question (GCD of consecutive integers is 1) by introducing distractions like negative exponents. That's not something you see in real GMAT questions. It also is a question where the statements are sufficient to give a 'no' answer, and such questions are very rare on the real GMAT - in most yes/no DS questions, if statements are sufficient, the answer is 'yes'. The basic concept in this question is difficult enough without it needing to be complicated further.
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