### IMO Shortlist 2013 problem N5

Kvaliteta:

Avg: 3,0Težina:

Avg: 8,0 Fix an integer . Two players, called Ana and Banana, play the following

An integer is called

Consider two integers with the property that each prime number divides if and only if it divides . Prove that either both and are good or both are bad.

*game of numbers*. Initially, some integer gets written on the blackboard. Then they make moves in turn, with Ana beginning. A player making a move erases the number just written on the blackboard and replaced it by some number with that is coprime to . The first player who cannot move anymore loses.An integer is called

*good*if Banana has a winning strategy when the initial number is , and*bad*otherwise.Consider two integers with the property that each prime number divides if and only if it divides . Prove that either both and are good or both are bad.

Izvor: Italy