## Chinese Remainder Theorem Part 2 – Non Coprime Moduli

As promised on the last post, today we are going to discuss the “Strong Form” of Chinese Remainder Theorem, i.e, what do we do when the moduli in the congruence equations are not pairwise coprime. The solution is quite similar to the one we have already discussed in the previous post, so hopefully, it will be a lot easier to understand.

### Prerequisite

If you haven’t read my previous post, you can find it here: Chinese Remainder Theorem Part 1. I strongly suggest you read that one before proceeding onwards.

# Problem

Given two sequences of numbers $A = [a_1, a_2, \dots, a_n]$ and $M = [m_1, m_2, \dots, m_n]$, find the smallest solution to the following linear congruence equations if it exists:

$$x \equiv a_1 \text{(mod } m_1) \\ x \equiv a_2 \text{(mod } m_2) \\ \dots \\ x \equiv a_n \text{(mod } m_n)$$

**Note**: The elements of $M$ are not pairwise coprime