Math & Physics Problems Wikia
Advertisement

Problem

今有物,不知其數。三、三數之,賸二;五、五數之,賸三;七、七數之,賸二。問:物幾何?

Suppose we have an unknown number of objects. When counted in threes, 2 are left over, when counted in fives, 3 are left over, and when counted in sevens, 2 are left over. How many objects are there?

[Assume the lowest positive integer solution]

Sunzige.gif

Solution

This problem is a system of indeterminate equations with infinitely many solutions.

Calculate the product of the moduli

.

The solution of the Chinese remainder theorem prescribes that

.

For this problem

where

are the modular inverses of each respective remainder, which can be solved systematically using the Euclidean algorithm. However, the numbers involved in this problem is small enough to be obtained by guessing and checking.

Calculation

Hence 233 is a solution. In fact, there are infinitely many solutions

where

is an integer. The lowest positive integer solution is

.

So there are 23 objects.

Advertisement