Question 54
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.*
*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.*