Theorem 1 (Chinese Remainder Theorem) Let $m_{1}, \cdots, m_{k}$ be pairwise coprime positive integers. Then, for any integers $a_{1}, \cdots, a_{k}$, the system of linear congruences
$$x \equiv a_{j}\left(\bmod m_{j}\right), \quad 1 \leqslant j \leqslant k$$
has a solution, and the solution is unique. In fact, let
$$c=M_{1} M_{1}^{-1} a_{1}+\cdots+M_{k} M_{k}^{-1} a_{k}$$

where $m=m_{1} \cdots m_{k}, m=m_{j} M_{j}(1 \leqslant j \leqslant k), M_{j}^{-1}$ is an integer satisfying
$$M_{j} M_{j}^{-1} \equiv 1\left(\bmod m_{j}\right), \quad 1 \leqslant j \leqslant k$$

(i.e., the inverse of $M_{j}$ modulo $m_{j}$) ${ }^{\oplus}$. Then, the solution to the system of congruences (3) is
$$x \equiv c(\bmod m)$$

Furthermore, $c$ is coprime to $m$ if and only if $a_{j}$ is coprime to $m_{j}$, $1 \leqslant j \leqslant k$.