Theorem

Suppose that $m_{1},\dots,m_{k}$ are mutually coprime integers and $a_{1},\dots,a_{k}$ are prescribed integers. Then, the system of congruences $$\begin{align*} x&\equiv a_{1}\pmod{m_{1}}\\ &\vdots\\ x&\equiv a_{k}\pmod{m_{k}} \end{align*}$$has a unique solution$\pmod{M}$ where $M=m_{1}\dots m_{k}$.