IMO 2026 - P1: The cute number theory

Problem Statement: There are $2026$ integers greater than $1$ written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m \ge 2$ and $n \ge 2$ from different places on the blackboard and replaces these two integers with \[\gcd(m,n) \quad \text{ and } \quad \frac{\mathrm{lcm}(m,n)}{\gcd(m,n)}.\] He continues to make moves while it is possible to do so. (a) Prove that, regardless of the choices of Confucius, after finitely many moves, exactly one integer $M$ on the blackboard is greater than $1$. (b) Prove that the value of $M$ does not depend on the choices of Confucius. (Note that $\gcd(x,y)$ denotes the greatest common divisor of positive integers $x$ and $y$, and $\mathrm{lcm}(x,y)$ denotes the least common multiple of $x$ and $y$.)