Odd Square Modulo 8

Theorem
Let $x \in \Z$ be an odd square.

Then $x \equiv 1 \pmod 8$.

Proof
Let $x \in \Z$ be an odd square.

Then $x = n^2$ where $n$ is also odd.

Thus $n$ can be expressed as $2 k + 1$ for some $k \in \Z$.

Hence:

But $k$ and $k + 1$ are of opposite parity and can therefore be expressed as $2 r$ and $2 s + 1$ (either way round).

Hence the result.

Also see

 * Square Modulo 8