BALTIC-WAY 2025 Problem 19

There are 100 positive integers written on a blackboard. Anna plays a game. A move consists of choosing two integers $a$ and $b$ on the blackboard with the property that $a \mid b$, erasing them both and writing the integer $b/a$. She makes moves until there are no more valid moves, at which point the game ends.

Find the largest $N$ such that for some initial state Anna can both:

• finish the game with only one number remaining, and
• finish the game with $N$ numbers remaining.