One application that I heard about (but haven't actually used myself) is in computing combinatorial numbers $x$ that have very many digits. Sometimes, instead of dealing with arbitrary precision arithmetic, it is easier to compute $x \pmod \mu$ for many small moduli $\mu$ (which can be substantially faster than dealing with $x$ itself) then afterwards combine the congruences using the Chinese Remainder Theorem to find $x$ itself. Although this requires that you have an upper bound on $x$ or some idea of how large it should be.