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, $$x^2 \equiv 1 \pmod 8$$.