(3) Chinese Remainder Theorem Let $m_{1}, m_{2}, \cdots, m_{k}$ be $k$ pairwise coprime positive integers, $M=$ $m_{1} m_{2} \cdots m_{k}, M_{i}=\frac{M}{m_{i}}(i=1,2, \cdots, k), b_{1}, b_{2}, \cdots, b_{k}$ be any integers, then the system of congruences
$$x \equiv b_{1}\left(\bmod m_{1}\right), \cdots, x \equiv b_{k}\left(\bmod m_{k}\right)$$

has a unique solution $x \equiv M_{1}^{*} M_{1} b_{1}+\cdots+M_{k}^{*} M_{k} b_{k}(\bmod M)$, where $M_{i}^{*}$ is any integer satisfying $M_{i}^{*} M_{i} \equiv$ $1\left(\bmod m_{i}\right)$ $(i=1,2, \cdots, k)$.