Any idea?

Let a1, a2, . . . , an Alice's numbers. Below, when X is a set of finite integers, we denote Q X the product of its elements. Moreover, we note Pi the set of prime factors of ai, and P the union of all sets Pi, i.e. the set of all prime numbers dividing at least one number ai. When Alice erases two numbers ai and aj to replace them with the numbers q = Q(Pi ∪ Pj ) and q2, she does not does not modify the set P, and replaces each of the sets Pi and Pj by their union Pi ∪ Pj . Thus, for all i, the set Pi can only grow (for inclusion) during the process, and since it is constrained to remain a subset of P, there is a moment from which Pi will no longer change. Selecting the latest of these moments as i varies, we obtain a moment, say m, from from which no more set Pi will change. We can even choose this moment so that any number ai which Alice one day acts upon has already been erased and replaced. Finally, we say that a number ai is small if it is equal to the product Q(Pi), and large if it is equal to Q(Pi)2; Otherwise, he is neutral. Each time a number ai undergoes an operation, it becomes small or large; thereafter it will be always either small or large. But then, from the moment m, the list of numbers written on the board will never change again. In indeed, if Alice erases two integers ai and aj , each of the integers ai and aj was already small or large. Since Pi = Pi ∪ Pj = Pj and ai̸ = aj , so one of the integers ai and aj is small, and the other large. But then the numbers q and q2 that Alice rewrites are equal to the integers ai and aj that she has just erased.

parmenides51 wrote: On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$. Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore. Note: The order of the numbers of the list is not important.