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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