MathDB

Let

m

boxes be given, with some balls in each box. Let

n < m

be a given integer. The following operation is performed: choose

n

of the boxes and put

1

ball in each of them. Prove:
(a) If

m

and

n

are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls.
(b) If

m

and

n

are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.

Problem Statement