Square Modulo 8

Theorem
Let $x \in \Z$ be an integer.


 * If $x$ is even then $x^2 \equiv 0 \pmod 8$ or $x^2 \equiv 4 \pmod 8$.
 * If $x$ is odd then $x^2 \equiv 1 \pmod 8$.

Proof for Even Integer
Let $x \in \Z$ be even.

Then from Square Modulo 4 $x^2 \equiv 0 \pmod 4$.

Hence there are two possibilities for $x^2$:
 * $x^2 \equiv 0 \pmod 8$;
 * $x^2 \equiv 4 \pmod 8$.

The fact that there do exist such squares can be demonstrated by example:
 * $2^2 = 4 \equiv 4 \pmod 8$;
 * $4^2 = 16 \equiv 0 \pmod 8$.

Proof for Odd Integer
Let $x \in \Z$ be odd.

Then from Odd Square Modulo 8, $x^2 \equiv 1 \pmod 8$.