Square Modulo 3

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

Then one of the following holds:

where:
 * $\divides$ denotes divisibility
 * $\nmid$ denotes non-divisibility.

Proof
Let $x$ be an integer.

Using Congruence of Powers throughout, we make use of:
 * $x \equiv y \pmod 3 \implies x^2 \equiv y^2 \pmod 3$

There are three cases to consider:


 * $(1): \quad x \equiv 0 \pmod 3$: we have $x^2 \equiv 0^2 \pmod 3 \equiv 0 \pmod 3$


 * $(2): \quad x \equiv 1 \pmod 3$: we have $x^2 \equiv 1^2 \pmod 3 \equiv 1 \pmod 3$


 * $(3): \quad x \equiv 2 \pmod 3$: we have $x^2 \equiv 2^2 \pmod 3 \equiv 1 \pmod 3$