In $\mathbb Z$: define $xRy \iff 3\mid(5x - 2y).$ Prove $R$ is an equivalence relation.
First I proved that it is reflexive, symmetric and transitive.
After that, I tried to find possible partitions of it ( $\prod$R ).
I started with $0$:
$$\begin{align} (0)&= \{ x \in \mathbb Z \mid xR0\} \\&= \{ x \in \mathbb Z \mid \exists k\in\mathbb Z(5x = 3k)\} \\&= \{ x \in \mathbb Z\mid \exists k(x=3k) \},\end{align}$$ since 3 and 5 are coprime then 3 divides x
Now I tried $1$:
$(1) = \{x \in \mathbb Z |xR1\} = \{x \in \mathbb Z \mid\exists k(5x = 3k + 1 )\}$ , how do I continue?
Is $\prod R = \{\{3k\mid k \in \mathbb Z \},\{3k+1\mid k \in \mathbb Z \},\{3k+2\mid k \in \mathbb Z \}\}$ ??