Do you mean the following (this is the general Chinese remainder theorem):

Let where . Then there exists auniqueso that for all .

Here are the steps:

1)Define .

2)Define , i.e. .

3)Prove that .

4)Therefore (use #3) the has unique solution in .

5)Denote solution in #4 by .

6)Define .

7)Prove that solves for each .

8)Now prove that there can be no other solution.