First of all, I have already proven that for every $d\in\mathbb{N}$ the relation
\begin{align*} a\sim_d b:\Leftrightarrow\exists k\in\mathbb{Z}:\,b-a=kd \end{align*}
is a congruence relation on $(\mathbb{Z},+,0,-)$. Furthermore, the trivial ones $\mathbb{Z}\times\mathbb{Z}$ as well as ${\rm id}_\mathbb{Z}$ are congruence relations.
Are those all of them and how do I prove it?