infiniti007 wrote:If 375y = x^2 and x and y are positive integers, then which of the following must be an integer?

I.) y/15

II.) y/30

III.) y^2/25

A.) I only
B.) III only
C.) I and II only
D.) I and III only
E.) I, II, and III only

Test the SMALLEST POSSIBLE CASE.

y = x²/375 = x²/(3*5*5*5).

To MINIMIZE the value of y, we must MINIMIZE the value of x.

Since y must be an integer, x² must be divisible by (3*5*5*5).
The least possible value of x such that x² is divisible by one 3 and three 5's is x=3*5*5, with the result that x² = (3*5*5)² = 3*3*5*5*5*5.

Plugging x² = 3*3*5*5*5*5 into the equation above, we get:
Least possible y = (3*3*5*5*5*5)/(3*5*5*5) = 15.

Implication:
y must be a MULTIPLE OF 15.

Test y=15 in the statements.

I: y/15 = 15/15 = 1.
II: y/30 = 15/30 = 1/2.
III: y²/25 = 15²/25 = (3*3*5*5)/25 = 9.

Statement II does not have to be an integer.
Eliminate C and E.

Since y=15 yields integer values in I and III, any greater multiple of 15 will also yield integer values in I and III.
Thus, I and III must both be integers.

The correct answer is D.

Another very fast approach is to find a value of y that works, then consider the consequences of the answers.

375y = a perfect square if y = 375.

Since y = 375 only implies I and III, we know the answer is A, B, or D.

Now notice that if y divides by 15, y² will divide by 225, so I's being true implies III's being true. So the answer CANNOT be A.

Now suppose the answer is III only. y² divides by 25 implies that y divides by 5. So I is false if y need not divide by 3. But 375 is not a perfect square: it only divides by 3, not by 9. So we need another factor of 3, meaning y must divide by 3.

Hence the answer is D, and we're done.

Under test constraints, you could probably skip every step after y = 375: you'd be better off picking with 70% certainty in 20 seconds than with 100% certainty in 3-5 minutes (or however long it could take to do the whole thing with assurance if you're not sure what to do at first).