Theorem (Chinese remainder 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 $M=m_{1}\dots m_{k}$.

Theorem (Generalized Chinese Remainder Theorem)

Suppose that $m_{1},\dots,m_{k}$ are 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 $M$ is the $$M=lcm(m_{1},\dots ,m_{k})$$if and only if $gcd(m_{i},m_{j})|a_{i}-a_{j},\forall i\not=j$.