Hi, there. I'm happy to provide a solution for this.

This one is very difficult, and not a likely GMAT problem. The more difficult problems on the GMAT still lend themselves to elegant solutions. This one has an irreducible plug-and-chug character to it.

Prompt:
If M, N, P, and Q are distinct digits, what is the value of Q?

A straightforward prompt. Not much to work with before we dive into the statements.

Statement #1: MNP * P = NQP, where MNP and NQP are three-digit numbers and M does not equal 0

Observation #1 -- the final digit P is a number such that its product with itself, its square, ends in the same digit. That is true of {1, 5, and 6}

P can't be 1, because MN1 * 1 would equal itself, not a new number.

Observation #2 -- If P = 5 or P = 6, the only way MNP * P will equal a three-digit product is if M = 1. If MNP is over 200, 5 or 6 times it would be more than three digits.

1N5*5 = NQ5 or 1N6*6 = NQ6

If P = 5, N must be greater than or equal to 6.

P = 5, N = 6, 165*5 = 825 (doesn't fit the pattern)
P = 5, N = 7, 175*5 = 875 (doesn't fit the pattern)
P = 5, N = 8, 185*5 = 925 (doesn't fit the pattern)
P = 5, N = 9, 195*5 = 975 ----> fits the pattern ---> Q = 7

If P = 6, N there's a problem --- the N of NQ6 must be large (7 or greater), because a number over 100 time 6, with all distinct digits, must be over 700. But if the N of 1N6 is that large, then 1N6*6 will be a four digit number. Therefore, P = 6 does not work with any cases that fit the pattern.

Statement #1 uniquely determines an answer to the question, so it is sufficient.

Statement #2: NQP > 500
With no information at all about M, and no information about N & Q & P except that they're all different, this statement is completely useless by itself. Statement #2 is insufficient.

Answer = A

Here's a DS question about digits more likely to appear on the GMAT.

https://gmat.magoosh.com/questions/1002

Let me know if anyone reading this has any questions about this.

Mike