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:

• $2^2 = 4 \equiv 4 \pmod 8$;
• $4^2 = 16 \equiv 0 \pmod 8$.

$\blacksquare$

Proof for Odd Integer

Let $x \in \Z$ be odd.

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

$\blacksquare$