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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.

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