Question 13
4a. (BUL 1) Let $m$ boxes be given, with some balls in each box. Let $n\lt m$ 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.
(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.