The article does not currently give an algorithm for finding solutions to the CRT problem. (From wikipedia, the attribution to Sun Zi is incomplete because he provides neither a proof nor algorithm. That article credits Aryabhata)
Also worth mentioning that factoring a number into prime powers is a good way to get pairwise coprime moduli: $n = p_1^{e_1} \cdots p_k^{e_k}$ For example, results mod 10^16 can be computed mod 2^16 and 5^16.

I am still investigating the full algorithm which I have never understood fully but like the extended Euclidean algorithm is based on Bezout's identity (which is quite fundamental for modular arithmetic)