Square Modulo 4

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


 * If $x$ is even then $x^2 \equiv 0 \pmod 4$.
 * If $x$ is odd then $x^2 \equiv 1 \pmod 4$.

Proof

 * Let $x \in \Z$ be even.

Then $\exists n \in \Z: x = 2 n$.

Hence $x^2 = \left({2n}\right)^2 = 4 n^2 \equiv 0 \pmod 4$.


 * Let $x \in \Z$ be odd.

Then $\exists n \in \Z: x = 2 n + 1$.

Hence $x^2 = \left({2n + 1}\right)^2 = 4 n^2 + 4n + 1 \equiv 1 \pmod 4$.