A Giant Blackboard

For any positive integer $n$ , we see that $\gcd(1, n) = 1$ . We see that, irrespective of the order in which we choose the numbers, the number $1$ cannot be removed from the blackboard. Every other number will be eliminated when it is paired with $1$ . Hence only $1$ remains at the end.

Relevant wiki: Greatest Common Divisor

Every time we choose the number 1, doesn't matter what number we choose next, because their greatest common divisor will be 1. Therefore, 1 wil be always on the board... thus, when there's only 2 numbers left, one of them is a 1, so the final number is still 1.