Mathematicians call this a system of congruences, and they write it like this: Our goal is to find a whole number, X, that has a remainder of 3 when divided by 5 and a remainder of 7 when divided by 8 (later we’ll add in the third condition, that it has a remainder of 2 when divided by 9). To better understand how the theorem works, let’s return to our example. “The Chinese remainder theorem gives you an actual recipe for making a number,”said Daniel Litt of the University of Georgia. Sun Tzu never proved this formally, but later the Indian mathematician and astronomer Aryabhata developed a process for solving any given instance of the theorem. The theorem allows you to find an unknown number if you know its remainders when it’s divided by certain numbers that are “pairwise coprime,” meaning they do not have any prime factors in common. What we do know is that the mathematical technique now known as the Chinese remainder theorem was devised sometime between the third and fifth centuries CE by the Chinese mathematician Sun Tzu (not to be confused with Sun Tzu who wrote The Art of War almost 1,000 years earlier). Legend suggests ancient Chinese generals actually employed this technique, though it’s unclear if they really did. At no point have you counted all your soldiers, but now you have enough information to determine the total number without having to state an explicit count that an enemy could intercept. Then you have them re-form in rows of eight, which leaves seven in the last row, and then rows of nine, which leaves two. You note that you end up with three soldiers in the last row. In a morning drill you ask your soldiers to line up in rows of five.

So you turn to a math trick that allows you to achieve both aims.

But you also need to know this information yourself. Imagine you’re a general in ancient times and you want to keep your troop counts secret from your enemies.

Ancient art of war 2