22. Let $m_{1}, m_{2}, \ldots, m_{k}$ be pairwise relatively prime positive integers. Let $M=m_{1} m_{2} \cdots m_{k}$ and $M_{j}=M / m_{j}$ for $j=1,2, \ldots, k$. Show that
$$M_{1} a_{1}+M_{2} a_{2}+\cdots+M_{k} a_{k}$$
runs through a complete system of residues modulo $M$ when $a_{1}, a_{2}, \ldots, a_{k}$ run through complete systems of residues modulo $m_{1}, m_{2}, \ldots, m_{k}$, respectively.