MathDB

Fix an integer

k>2

. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer

n \ge k

gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number

m

just written on the blackboard and replaces it by some number

m'

with

k \le m' < m

that is coprime to

m

. The first player who cannot move anymore loses.
An integer

n \ge k

is called good if Banana has a winning strategy when the initial number is

n

, and bad otherwise.
Consider two integers

n,n' \ge k

with the property that each prime number

p \le k

divides

n

if and only if it divides

n'

. Prove that either both

n

and

n'

are good or both are bad.

Problem Statement