Do you mean the following (this is the general Chinese remainder theorem):
Let where . Then there exists a unique so that for all .
Here are the steps:
2)Define , i.e. .
3)Prove that .
4)Therefore (use #3) the has unique solution in .
5)Denote solution in #4 by .
7)Prove that solves for each .
8)Now prove that there can be no other solution.