ELMO 2022 Problem 1

Let $n > 1$ be an integer. The numbers $1, 2, \ldots, n$ are written on a board. Aliceurill and Bobasaur take turns circling an uncircled number on the board, with Aliceurill going first. When the product of the circled numbers becomes a multiple of $n$ , the game ends and the last player to have circled a number loses. For which values of $n$ can Bobasaur guarantee victory?

Max Lu