IMO Shortlist 2013 problem N5

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Dodao/la: arhiva
21. rujna 2014.
Fix an integer k \geq 2. Two players, called Ana and Banana, play the following game of numbers. 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 good if Banana has a winning strategy when the initial number is n, and bad 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.
Izvor: Italy