|
-
Problem 1. There are 2026 integers greater than 1 written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m > 1$ and $n > 1$ from different places on the blackboard and replaces these two integers with
$$\gcd(m, n) \quad \text{and} \quad \frac{\text{lcm}(m, n)}{\gcd(m, n)}.$$
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 $\text{lcm}(x, y)$ denotes the least common multiple of $x$ and $y$.)
Solution 1Solution 2Solution 3Solution 4Solution 5
|
