Cube Modulo 9

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

Then one of the following holds:

Proof
Let $x$ be an integer.

There are three cases to consider:


 * $(1): \quad x \equiv 0 \pmod 3$: we have $x = 3 k$


 * $(2): \quad x \equiv 1 \pmod 3$: we have $x = 3 k + 1$


 * $(3): \quad x \equiv 2 \pmod 3$: we have $x = 3 k + 2$

Then: