Han Xin ordered soldiers, also known as Chinese remainder theorem. According to legend, Emperor Gaozu Liu Bang asked General Han Xin how many soldiers he commanded. Han Xin replied that there were 1, 5, 2, 7, 4, 13 soldiers in a row for every three people ... Liu Bang was at a loss and didn't know the number.

Let's consider the following questions first: Suppose the number of soldiers is less than 10,000, and there are only three people left for every five, 13, 17, so how many soldiers are there?

First find the least common multiple of 5,9, 13 and 17 (note: because 5,9, 13 and 17 are pairwise coprime integers, the least common multiple is the product of these numbers), and then add 3 to get 9948 (person).

There is a similar question in China's ancient mathematical work Sun Tzu's Art of War: "There are things today, I don't know their numbers, three or three numbers, two, five or five numbers, three or seven numbers, two, ask about the geometry of things? 」

A: "Twenty-three"

The technique said, "If the number of three plus three leaves two, it will be one hundred and forty, five plus five plus three, sixty-three, seven plus seven plus two, thirty, and if the number of three plus three leaves one, it will be seventy, five plus five plus one, and it will be fifteen."

It is impossible to verify the author and the exact date of the book, but according to the research, the date of the book will not be after the Jin Dynasty. According to this research, the solution of the above problem was found earlier in China than in the west, so the generalization of this problem and its solution are called China's remainder theorem. China's remainder theorem plays a very important role in modern abstract algebra.