We know that odd x odd = odd. Here's a quick proof:
Say the odd number are a = 2x + 1 and b = 2y + 1. Then, we have
ab = (2x + 1)(2y + 1) = 2y(2x + 1) + 2x + 1 => 2(2xy + y) + 2x + 1 = even + odd = odd.
We know 5^23 = 5 * 5 * 5 * 5 * ..... * 5, with 23 5's. The first two fives pair up into an odd, and so on, so that product is odd. Repeat for 7^17 to get two odd numbers, and we know that odd + odd = even, so the answer is 2.