Math story #1 - Han Xin Counting Soldiers (韓信點兵) (Remainder problem)

Han Xin (韓信) (died 196 BC) was a military general who served Liu Bang during the Chu–Han Contention and contributed greatly to the founding of the Han dynasty.

In one of his many battles, he brought 1500 soldiers with him. It was a tough battle, about a little more than four hundred soldiers were killed. When they returned to their headquarters, general Han needed to know how many soldiers survived.

However, most of them could hardly count past ten.
Then he ordered the soldiers to line up 3 in a row, and the last row remains 2 soldiers.
Then he ordered them to line up 5 in a row, and the last row remains 4 soldiers.
Then he ordered them to line up 7 in a row, and the last row remains 6 soldiers.
Knowing those numbers, he quickly calculated how many soldiers he still had.

Actually the problem is related to Indeterminate equations of Number theory.

x = 2 (mod 3) (It means x-2 can be divided by 3)
x = 4 (mod 5)
x = 6 (mod 7)

Can you figure it out?

Solution:

Let's x be the number of soldiers. In the above case, it is easier since x+1 is divisible by 3, 5, and 7,
that is, x+1 = n(3)(5)(7) = 105n, where n is a positive integer.
Since a little more than 400 soldiers were killed, the number of surviving soldiers is around 1000 to 1100.
Therefore, n has to be 10; and x = (105)(10)-1 = 1049.

Later, the similar problem was written in Sunzi Suanji (孫子算經) which was a famous mathematical treatisewritten during 3rd to 5th centuries.