Chinese Remainder Theorem

Let be positive integers which are pairwise coprime. Let .

Theorem

Let . Then there exists a solution to the system of simultaneous congruences . Moreover this solution is unique modulo .

Proof

Uniqueness is easy.
Define .
Then so we can find and also for any .
Then is a solution

Theorem

The map:

is a ring isomorphism.

Proof

We just need to check that its bijective, but this is exactly the previous theorem.

Corollary

There is a group isomorphism

Proof

Just look at components, its fine.