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I didn't read this obvious application yet in the other answers: if you want to solve a polynomial equation in $\Bbb{Z}/n\Bbb{Z}$ and you factorize $n=q_1\dots q_m$ via CRT, you can calculate the solution set $S_i$ in each $\Bbb{Z}/q_i\Bbb{Z}$ and the total solution set is then $$S_1×\dots ×S_m \subset\Bbb{Z}/q_1\Bbb{Z}×\dots ×\Bbb{Z}/q_m\Bbb{Z}$$.