MathDB

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.

Problem Statement