ISL 2013 N5

Fix an integer $k \geqslant 2$. Two players, called Ana and Banana, play the following game of numbers: Initially, some integer $n \geqslant k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m^{\prime}$ with $k \leqslant m^{\prime}<m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \geqslant k$ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n, n^{\prime} \geqslant k$ with the property that each prime number $p \leqslant k$ divides $n$ if and only if it divides $n^{\prime}$. Prove that either both $n$ and $n^{\prime}$ are good or both are bad. (Italy)