Question

Are perfect squares only congruent to 0mod4 or 1mod4?

A) 0mod4
B) 1mod4
C) Both 0mod4 and 1mod4
D) Neither 0mod4 nor 1mod4

Answer 1

Perfect squares can be congruent to either 0 mod 4 or 1 mod 4.

Step-by-step explanation:

Perfect squares are numbers that are obtained by squaring an integer. When a perfect square is divided by 4, there are only two possible remainders: 0 and 1.

Let's consider some examples:

When we square any even number, the result is always divisible by 4. For example, 2^2 = 4, 4^2 = 16, 6^2 = 36, and so on. In these cases, the remainders are 0 when divided by 4, so they are congruent to 0 mod 4.

When we square any odd number, the result is always of the form 4n + 1, where n is an integer. For example, 3^2 = 9 (which is 4 * 2 + 1), 5^2 = 25 (which is 4 * 6 + 1), 7^2 = 49 (which is 4 * 12 + 1), and so on. In these cases, the remainders are 1 when divided by 4, so they are congruent to 1 mod 4.

So, perfect squares are congruent to either 0 mod 4 or 1 mod 4.