This is a flawed question.
Is y!/x! an integer?First of all,
factorial is defined only for non-negative integers, so realistic GMAT question would mention that
x and
y are non-negative integers.
Next,
\frac{y!}{x!}=integer will hold true if
y\geq{x}. So, the question basically asks whether
y\geq{x}.
(1) (x + y)(x-y) = 5! + 1 -->
x^2-y^2=121. As discussed, since
x and
y must be non-negative integers, then
x>y and the asnwer to the question is NO. Sufficient.
(2) x + y = 112. Not sufficient to answer whether
y\geq{x}.
Answer: A.
Now, even though formal answer to the question is A, this is not a realistic GMAT question, as:
on the GMAT, two data sufficiency statements always provide TRUE information and these statements never contradict each other. But from (1) the only non-negative integer solutions for
x and
y are: (11, 0) and (61, 60), so
x+y cannot equal to 112 as the second statement says, which means that the statements clearly contradict each other.
The question is flawed. You won't see such a question on the GMAT.
Hope it's clear.
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