Quickies
Alice and Bob play the following game: they first agree on a positive integer n > 1, called the target number, which remains fixed, and each turn, a player names a prime factor of n, and the state of the game is multiplied by this prime.
Abstract
Alice and Bob play the following game. They first agree on a positive integer n > 1, called the target number, which remains fixed. Throughout, the state of the game is represented by an integer, which is initially 1. Players take turns with Alice going first. Each turn, a player names a prime factor of n, and the state of the game is multiplied by this prime. If a player manages to make the state of the game exactly equal to the target number, n, then that player wins. If the state of the game ever exceeds n, then the game is declared a draw. Determine for which values of n each player has a winning strategy, and for which values of n the game should end in a draw.