With (1) we know that x^2 - y^2 = 5! +1
Is y!/x! an integer?
Well, that's a factorial divided by a factorial. We know that 0! = 1.
And since x^2 - y^2 = some positive integer, we know that x and y are going to be integers as well (no decimals, a 7.13! would be weird!)
So...x and y are integers. y! / x! ...okay we well can get into dangerous territory if the denominator is a large number --- that introduces fractions. If y>x, then numerator will always be larger and we won't necessarily get that fraction problem.
But in this case, we know that x^2 - y^2 = some positive integer...
... so x>y. That means we know DEFINITIVELY the denominator is going to be bigger than the numerator and we know DEFINITIVELY that we get a fraction less than one.
In that case, is y! / x! an integer? Well we know definitively that it's a fraction less than 1, thus not an integer. Thus, we have enough information to say whether YES integer or NO, not an integer. The answer would be NO not an integer. But don't confuse that with the actual question it's asking...it's not do we have an integer yes or no? THe question is asking...do we have SUFFICIENT INFORMATION to determine whether the answer to that question is yes or no. And the answer is we DO have SUFFICIENT INFORMATION to make a definitive yes or no answer.
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