MathNet 2023-part-02 — 191 problems (EN–VI) Question 73

Alice has arranged 200 boxes in her living room. Each box contains a piece of paper on which she has written a positive integer; the 200 integers are not necessarily distinct. Every minute, as long as possible, Alice performs an action of the following form: she chooses three boxes containing integers $a, b$ and $c$ such that $a+b=c$, and also chooses an arbitrary integer $k \geqslant 2$; she then replaces the integer $c$ by the integer $k \times c$. If she can no longer perform such an action, she stops for good.
Prove that, whatever the initial situation and Alice's choices, she will be forced to stop at some point.