10. Show that the solutions to the simultaneous system of congruences
$$\begin{aligned} x & \equiv a_{1}\left(\bmod m_{1}\right) \\ x & \equiv a_{2}\left(\bmod m_{2}\right) \\ & \\ x & \equiv a_{r}\left(\bmod m_{r}\right) \end{aligned}$$
where the $m_{j}$ are pairwise relatively prime, are given by
$$x \equiv a_{1} M_{1}^{\phi\left(m_{1}\right)}+a_{2} M_{2}^{\phi\left(m_{2}\right)}+\cdots+a_{r} M_{r}^{\phi(m)}(\bmod M)$$
where $M=m_{1} m_{2} \cdots m_{r}$ and $M_{j}=M / m_{j}$ for $j=1,2, \ldots, r$.