The answer is actually pretty easy. We are looking for an integer that divides 15! + 13, so (15! + 13) / q = n for some integers q and n. Using properties of fractions, this means (15!/q)+ (13/q) = n. Since 13 is prime, we know q must be 1 or 13. 1 is not an answer choice, so q must be 13. This can be checked further because by noting that 15!=15x14x13x12..., so we see 13 divides 15!, because 13 divides the third term of the factorial. Therefore 13 divides 15! and13, so 13 divides 15!+13 based on the above property of fractions.
The solution factoring out 13 is a better solution though.