Let x be an integer. Prove that x^2 is even if and only if is divisible by 4.

Assume that x^2 is even. By definition, this means that x^2 = 2k for some integer k. Now, let's express x in terms of its prime factorization: x^2 = (2m)^2 for some integer m (since x^2 is even, we can write it as 2 times another integer). Simplifying further, we get x^2 = 4m^2. This implies that x^2 is divisible by 4. Therefore, if x^2 is even, then x is divisible by 4