Final answer:
Perfect squares are congruent to 0 mod 4 or 1 mod 4 because the squares of even numbers are divisible by 4 and the squares of odd numbers have a remainder of 1 when divided by 4.
Step-by-step explanation:
Numbers that are perfect squares are only congruent to 0 mod 4 or 1 mod 4 because when you square a number, depending on whether it is even or odd, the square will have a specific form. An even number can be written as 2k, and when squared, it becomes (2k)² = 4k², which is clearly divisible by 4, hence congruent to 0 mod 4. An odd number can be represented as 2k+1, and when squared, we have (2k+1)² = 4k² + 4k + 1, which is of the form 4m + 1 (where m is an integer), thus it is congruent to 1 mod 4.