Problem 1

There are 2026 integers greater than 1 written on a blackboard, not necessarily different. In one move, Confucius chooses two integers (m>1) and (n>1) from different positions on the blackboard and replaces them by

[ \gcd(m,n) \qquad\text{and}\qquad \frac{\operatorname{lcm}(m,n)}{\gcd(m,n)}. ]

He continues making moves for as long as this is possible.

Prove that, regardless of Confucius's choices, after finitely many moves exactly one integer (M) on the blackboard is greater than 1.
Prove that the value of (M) does not depend on Confucius's choices.

Here (\gcd(x,y)) denotes the greatest common divisor of positive integers (x,y), and (\operatorname{lcm}(x,y)) denotes their least common multiple.