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]
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
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.
Hence 233 is a solution. In fact, there are infinitely many solutions
is an integer. The lowest positive integer solution is
So there are 23 objects.