Pohlig–Hellman discrete logarithm computation is based on the CRT.
A degree n discrete Fourier Transform is identical to polynomial evaluation at primitive $n$-th roots of unity. The inverse transform is interpolation or the CRT (as mentioned in earlier posts).
Montgomery reduction uses the CRT to reduce an integer modulo $x\bmod{N}$ without division by $N$ by creating an integer equivalent to $x\bmod{N}$ and $0\bmod{2^r}$, then dividing by $2^r$.