Školjka

%V0 Fix an integer $k \geq 2$. Two players, called Ana and Banana, play the following [i]game of numbers[/i]. Initially, some integer $n \geq k$ gets written on the blackboard. Then they make moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaced it by some number $m'$ with $k \leq m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \geq k$ is called [i]good[/i] if Banana has a winning strategy when the initial number is $n$, and [i]bad[/i] otherwise. Consider two integers $n, n' \geq k$ with the property that each prime number $p \leq k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.