FI-MO 2017 Problem 4

Let $m$ be a positive integer. Two players, Axel and Elina play the game HAUKKU ( $m$ ) proceeds as follows: Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$ , loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ( $m$ ) game for all $m \in Z_{+}$ .

PS. As member Loppukilpailija noted, it should be written $m>1$ , as the statement does not hold for $m = 1$ .