GMAT Challenge Question: Divisibility

One of the most common concepts tested on the GMAT quantitative section is divisibility, and the data sufficiency problem below is a prime example of how the exam can ask you to use your factoring/divisibility skills.  Take a shot at this question, post your answers (and show your work!) in the comments field, and check back later for a full solution.

For positive integers x and y, x^2 = 350y.  Is y divisible by 28?

(1)    x is divisible by 4

(2)    x^2 is divisible by 28

UPDATE: Solution!

Solution: A.  In order to be divisible by 28, a number must have the prime factors 2*2*7.  From the given information we can conclude that x^2 = 2*5*5*7*y  (the factors indicate the prime factorization of 350), and that, because we need the factors to come in pairs in order for the square root of x^2 to be an integer, y is then divisible by at least 2 and 7.  From that, we need the statements to supply one more factor of 2 for y.  Statement 1 does exactly that: because x is divisible by 4, then x^2  will be divisible by 4*4, or 2*2*2*2.  As 350 only has one 2, y must supply the rest, so we can conclude that y is divisible by 2*2*2*7.  As only 2*2*7 was required for y to  be divisible by 28, statement 1 is sufficient.  Statement 2 only tells us that y is divisible by 2*7 – and we already knew that. Accordingly, statement 2 is not sufficient, and the correct answer is A.