Just to follow up on my last post, that solution relies on multiplicative remainders, a topic that shouldn't appear on the GMAT. To get a feel for how these work, consider something like this:
5 has remainder 1 when divided by 4. (For the rest of the post, I'll use the term "mod 4" rather than "when divided by 4".) Notice that 5 * 5 = 25, which also has remainder 1 mod 4. This because the remainders are multiplicative: if we multiply remainder 1 mod 4 by remainder 1 mod 4, we end up with remainder 1 * 1 mod 4. We need to be treating each number by the same mod (we can't do remainder 1 mod 4 * remainder 1 mod 3, at least not that simply), but if we do, the multiplication sorts out our remainders for us.
With that in mind, once I see that 49 = 1 mod 24, I can replace any 49 I see with 1, so 49 * 49 * 49 * ... * 49 is the same, mod 24, as 1 * 1 * 1 * ... * 1.
