Problem of the Week 1235

A Prime Multiplication Game

Alice and Bob play a game. The target is an integer N at least 2. The state of the game, M, is an integer that starts out being 1. Players alternate moves, with Alice going first.

At each move, a player multiplies M by a prime number that divides N. A player wins by making M equal to N. If M ever exceeds N, the game is a draw.

For which values of N does Alice have a winning strategy? For which values of N does Bob have a winning strategy?

Source: Math. Magazine Problem 1985 posed by Brooke Tooles, a Rutgers student; Feb. 2017, 90:1, p. 80-81.

[View the solution]