I need to find equivalence classes of relation $\sim$ \begin{array}{l} x\sim y \Leftrightarrow 4 \mid\left(x^{3}-y^{3}\right) \\ \text { for } x, y \in \mathbb{Z} \end{array}
I have shown that $\sim$ is equivalence relation. I blind guessed that all even numbers will be one equivalence class, so my question is is there some systematic approach to this problem? Any help is appreciated :)
We have $x\sim y \iff $ $x^{3}$ and $y^{3}$ leave the same remainder when divided by $4$.
Write $x=4q+r$ with $r\in\{0,1,2,3\}$. Then $x \sim r$ (check!) and so it suffices to consider the equivalence class of $r$. In particular, there are at most four equivalence classes. In fact, it turns out that there are three equivalence classes: $[0]$, $[1]$, $[3]$, because $[2]=[0]$.
In other words, all even numbers are in the same class, but the odd numbers fall into two classes.