If x and y are positive integers, is x y an integer?

VERITAS PREP OFFICIAL SOLUTION:

Question Type: Yes/No. The question asks: “Is x/y an integer?”

Given information in the question stem or diagram: x and y are positive integers.

Statement 1: Every factor of y is also a factor of x. This means that whatever factors are in the denominator, those same factors are also in the numerator. So if y = 30 (with factors of 2 • 3 • 5) then x will have at least these factors. In other words, x will be a multiple of 30. If x has all of the factors of y then x/y will always be an integer. This statement is sufficient, and the answer is either A or D.

Statement 2: Every factor of x is also a factor of y. This statement may seem to be identical to Statement 1, but it is not! All of the factors of x will also be present in y, but y could contain other factors. For instance, x could be 30 and y could be 60, meeting all the conditions in this statement. Or they could each be equal to 30. This statement allows for x/y to be the integer 1 but also many non-integers, so it is not sufficient and the correct answer is A. Note: In any question like this, where the statements appear to be the same, you should be highly suspicious of choice D as the correct answer!

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