Parity of Integer equals Parity of its Square/Even

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

Let $p$ be even.

Then $p^2$ is also even.

Proof
Let $p$ be an integer.

By the Division Theorem, there exist 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.

Let $r = 0$, so:
 * $p = 2 k$

Then:
 * $p^2 = \paren {2 k}^2 = 4 k^2 = 2 \paren {2 k^2}$

and so $p^2$ is even.