# Positive integers on a blackboard

Consider a blackboard with some positive integers written on it.

A move consists one of the following actions:

1. Choose two integers $m$ and $n$ on the board, remove them, and write $m+n$ on the board.
2. 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?

• Can $m$ be prime for the 2nd type of move? Jul 22 '16 at 11:08
• Does your rule with distinct prime factors mean that for example for a 4 you only write a single 2 back to the board? Jul 22 '16 at 11:08
• @TheDarkTruth Yes. Jul 22 '16 at 11:08
• @Ankoganit No, I will add that. Jul 22 '16 at 11:09
• No. 1 is not a prime. Jul 22 '16 at 11:09

Suppose it can continue infinitely long. Note that the $2$nd type of operation strictly decreases the sum of numbers on the board, whereas the $1$st operation leaves it invariant. So we can perform the $2$nd operation only finitely many times. Consider the last occurrence of the 2nd operation. Some there can't be infinitely many operation of there first type between two consecutive operations of the first type, we have made only finitely many moves so far. (Thanks to elias for this part). After the last occurrence of the $2$nd operation, we can only make the first type of operation, which always decreases the number of numbers on the board by one. So it has to stop eventually. $\mathrm{QED}$