Let $m\in \mathbb{Z}$ and consider the relation $R_m$ defined by $a{R}_{m}b$ if and only if $a\equiv b(\mathrm{mod}m)$. Then $R_m$ is:

$a\equiv b(\mathrm{mod}m)$ means $m$ divides $(a-b)$.Reflexive: $a-a=0$. Since $m$ divides $0$ for any $m\ne 0$, $a\equiv a(\mathrm{mod}m)$ holds. $R_m$ is reflexive.Symmetric: If $a\equiv b(\mathrm{mod}m)$, then $m$ divides $(a-b)$. This implies $m$ divides $-(a-b)=(b-a)$, so $b\equiv a(\mathrm{mod}m)$. $R_m$ is symmetric.Transitive: If $a\equiv b(\mathrm{mod}m)$ and $b\equiv c(\mathrm{mod}m)$, then $m|(a-b)$ and $m|(b-c)$.Summing them: $m|((a-b)+(b-c))⟹m|(a-c)$, so $a\equiv c(\mathrm{mod}m)$. $R_m$ is transitive.Since the relation is reflexive, symmetric, and transitive, it is an equivalence relation.