One answer I don't see here: Lagrange interpolation. If one takes, for example, the ring $\\mathbb{Q}[x]$, and realizes CRT as a statement about rings and direct sums of $R/P$ over a set of co-prime $P,$ then one can construct polynomials which have cycles of arbitrary length in the rationals (or any number of cycles of arbitrary length). Lagrange interpolation has other applications, but the proof is CRT.

One answer I don't see here: Lagrange interpolation. If one takes, for example, the ring $\mathbb{Q}[x]$, and realizes CRT as a statement about rings and direct sums of $R/P$ over a set of co-prime $P,$ then one can construct polynomials which have cycles of arbitrary length in the rationals (or any number of cycles of arbitrary length). Lagrange interpolation has other applications, but the proof is CRT.