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

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:

$2^2 = 4 \equiv 4 \pmod 8$

$4^2 = 16 \equiv 0 \pmod 8$

$\Box$

Proof for Odd Integer

Let $x \in \Z$ be odd.

Then from Odd Square Modulo 8:

$x^2 \equiv 1 \pmod 8$

$\blacksquare$