ÖMO 2018 Problem 3

There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$ , Round $2$ , etc., additional pieces of candy are distributed among the children according to the following rule: In Round $k$ , each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece. Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy.

(Proposed by Theresia Eisenkölbl)