Fix an integer k>2. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer n≥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′ with k≤m′<m that is coprime to m. The first player who cannot move anymore loses.An integer n≥k is called good if Banana has a winning strategy when the initial number is n, and bad otherwise.Consider two integers n,n′≥k with the property that each prime number p≤k divides n if and only if it divides n′. Prove that either both n and n′ are good or both are bad. inductionnumber theorygameIMO Shortlist