KöMaL 2024 A879

An integer $k>2$ is given. Xavier and Yvette play the following game: a number $n>k$ is initially written on the blackboard. The two players take turns, Xavier starts. In each turn the integer $m$ on the blackboard is replaced by integer $m'$ satisfying $k\le m'<m$ and $\gcd (m,m')=1$. The player who cannot make a legal move loses the game. We say that integer $n>k$ is good if Yvette has a winning strategy. Prove that if $n$, $n'>k$ are two integers satisfying the condition that every prime $p\le k$ divides $n$ if and only if it divides $n'$, then $n$ is good if and only if $n'$ is good.