If x and k are both integers, x > k, and x−k = 625, what is x? (1) |k|

Interesting problem!

The first thing I noticed is that the question itself really limits the possibilities. There can't be that many integers where \(x^{−k} = 625\), right? Also, I know 625 as a 'special number' - you should memorize the perfect squares up to about 25^2, so that you notice things like this quickly on test day. 625 = 25^2, so I immediately know one of the possibilities. x could be 25, and k could be -2. (Note the 'double negative' there).

However, you should never be able to solve a DS question without either statement. That's something that never happens on DS. So, there must be at least one possibility. The GMAT likes to trick you into forgetting about the simplest exponent of all: 1. x could be 625, and k could be -1.

Also, notice that 25 can be factored down more. 25^2 = 5^4. So, finally, x could be 5, and k could be -4.

List the three possibilities on your paper:
x = 25, k = -2
x = 625, k = -1
x = 5, k = -4

Then, start working with the statements. Your question to ask yourself: do the statements let me 'narrow it down' to just one of these possibilities?

(1) does exactly that. 2 is the only prime value for k in our list. So, if we know that |k| is prime, then the first possibility is the only one that works. (1) is sufficient.

(2) is insufficient, because the first two possibilities could both work. (They're really hoping that you don't think of x = 625, k=-1. If you didn't think of that, you'd think this was sufficient as well.)