If p is a positive integer, what is the remainder when p2-1 is divide

p-1, p, and p+1 are three consecutive integers. Since multiples of 3 are three apart, among any three consecutive integers, we must have exactly one multiple of 3. If, as each Statement tells us, p is not a multiple of 3, then one of p-1 or p+1 will be.

If p is odd, as each Statement tells us, then p-1 and p+1 are consecutive even integers. Every second even integer is a multiple of 4, so exactly one of p-1 or p+1 is a multiple of 4 (at least), and the other is a multiple of 2.

p^2 - 1 = (p + 1)(p - 1), and since we can be sure we have a multiple of 3, a multiple of 4, and another multiple of 2 in the product (p+1)(p - 1), this product must be divisible by (3)(4)(2) = 24, and we'll get a remainder of zero when we divide it by 24. So the answer is D.

There's an older GMATPrep question that is very similar to this one.