According to the definition, the residue classes have the following properties:
(1) $\mathbf{Z}=k_{0} \cup k_{1} \cup k_{2} \cup \cdots \cup k_{m-1}$, and $k_{i} \cap k_{j}=\varnothing(i \neq j)$;
(2) For $\forall n \in \mathbf{Z}$, there is a unique $r_{0} \in\{0,1,2, \cdots, m-1\}$, such that $n \in k_{r_{0}}$;
(3) For $\forall a, b \in \mathbf{Z}, a, b \in k_{r} \Leftrightarrow a \equiv b(\bmod m)$.