Show that every perfect square is congruent to 0, 1, or 4 modulo 8

You want to prove that



for (presumably) all integers .

Let's consider some sub-cases.

Suppose is odd. Then



If is even, then so is , which means you can write for some integer , and this reduces to .

If is odd, the same thing happens; you get that is still even, so and you're left again with .

Now assume is even. Then . If is even, then you will always be able to write



Meanwhile, if is odd, then



So you conclude that

The answer is b) 32 your welcome