Introduction

The classic version of Chinese remainder theorem tells us that we can find solutions out of modulus relations. You may have seen this poem when you were young.

有物不知其數，三三數之剩二，五五數之剩三，七七數之剩二。問物幾何？

Translation:

There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there?

This poem can be translated into finding the solution of an equation:

$$\begin{cases}x \equiv 2 \pmod 3, \\x \equiv 3 \pmod 5, \\x \equiv 2 \pmod 7.\end{cases}$$

In modern language...