Parallel computation: Suppose you have a huge computation to do that involves adding, multiplying and subtracting integers. Possibly also dividing but, if so, only division by numbers in a finite set S which you already know.

Choose primes $p_1$, $p_2$, ..., $p_r$ which do not divide any element of $S$, and such that $p_1 p_2 \cdots p_r$ is surely larger than your answer. Split your computation over $r$ processors, the $i$th of which computes the answer modulo $p_i$. Use CRT to put your answer back together in the end.

This was the method used in the recent computation of the Kazhdan-Lustig-Vogan polynomials of E8.