Parity of Integer equals Parity of its Square

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

Then $p$ is even iff $p^2$ is even.

Proof
Let $p$ be an integer.

By the Division Theorem, there are unique integers $k$ and $r$ such that $p = 2k + r$ and $0 \le r < 2$.

That is, $r = 0$ or $r = 1$, where $r = 0$ corresponds to the case of $p$ being even and $r = 1$ corresponds to the case of $p$ being odd.