Question 1
There are 2026 integers greater than 1 written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m\gt 1$ and $n\gt 1$ from different places on the blackboard and replaces these two integers with
$$\gcd(m,n)\quad\text{and}\quad \frac{\operatorname{lcm}(m,n)}{\gcd(m,n)}.$$
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.
Here $\gcd(x,y)$ denotes the greatest common divisor and $\operatorname{lcm}(x,y)$ the least common multiple of positive integers $x,y$.
$$\gcd(m,n)\quad\text{and}\quad \frac{\operatorname{lcm}(m,n)}{\gcd(m,n)}.$$
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.
Here $\gcd(x,y)$ denotes the greatest common divisor and $\operatorname{lcm}(x,y)$ the least common multiple of positive integers $x,y$.