Consider a blackboard with some positive integers written on it.
A move consists one of the following actions:
- Choose two integers $m$ and $n$ on the board, remove them, and write $m+n$ on the board.
- Choose one composite integer $m$ on the board, remove it and write all distinct prime factors of $m$ on the board.
The game ends when there is only one prime number on the board, or just the number one.
Does the game always end?