Britannica Sources used in this answer

How did Qin Jiushao develop the Chinese remainder theorem?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

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.

In his 1247 work, Shushu jiuzhang, Qin Jiushao provided the first complete description of the Chinese remainder theorem, building upon obscure methods previously mentioned in a 5th-century treatise by Sun Zi.12

Qin developed this algorithm primarily to clarify procedures used in calendrical computation that astronomers were applying without full understanding. His approach involved:1

Relatively Prime Moduli: For cases where moduli are relatively prime, he solved the congruence $x \cdot a \equiv 1 (\mod b)$ by using a sequence of quotients derived from the greatest common divisor of $a$ and $b$ , which corresponds to the continued fraction for $b / a$ .1
Generalization: He addressed cases where moduli are not relatively prime by first eliminating their common factors to reduce the problem to the relatively prime case.1

While Qin’s book is the earliest surviving text to detail this process, the historical development between Sun Zi and Qin remains unknown.1