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How can the remainder problem posed by Sun Zi be solved?

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East Asian mathematics Counting boards and markers, or counting rods, were used in China to solve systems of linear equations. This is an example from the 1st century ce.

The remainder problem posed by the 3rd-century mathematician Sun Zi involves finding an integer that satisfies a system of simultaneous congruences based on different divisors (moduli). While Sun Zi provided specific examples—such as finding a number that leaves remainders of 2, 3, and 2 when divided by 3, 5, and 7, respectively—the original method he used remains obscure.123

A general, systematic procedure for solving these problems was not fully articulated until 1247 by Qin Jiushao, whose work is now known as the Chinese remainder theorem.123

To solve such a system:
* Identify Moduli: Ensure the moduli are pairwise relatively prime. If they are not, common factors must be eliminated first.13
* Apply Algorithms: Qin Jiushao developed an algorithm to find the modular inverse, solving for $x$ in the congruence $x a \equiv 1 (\mod b)$ .3
* Calculate: By utilizing a sequence of quotients similar to the process for finding the greatest common divisor or continued fractions, one can determine the unique solution modulo the product of the divisors.13