Modular Arithmetic CRT: How do modulo with very big numbers

I have always been intrigued as to how one would calculate the modulo of a very large number without a calculator. This is an example that I have come up with just now:

4239^4 mod 19043

The answer is 808, but that is only because I used a calculator. I read in books and online that you can break the modulo 19043 to its factors such that it is modulo 137 and 139 as (modulo (137*139)) is (modulo 19043).

I tried something like this...

4239^4 mod 137
=129^4 mod 137
=123


4239^4 mod 139
=69^4 mod 139
=113

But now I am stuck as to what to do next in Chinese Remainder Theorem