Question 53
Anu and Bert each play the following game with ChatGPT. Anu's game starts with the number $2023!$ on the board, Bert's game with $2024!$ on the board. Each move consists of two parts. First the active player divides the number on the board by one of its composite factors $d$. Then their opponent must do one of the following actions:
(1) multiply the number on the board by any factor $d'$ of $d$ satisfying $1 \lt d' \lt d$;
(2) multiply the number on the board by $7$;
(3) divide the number on the board by $10$ and multiply the result by $2023$ (this option cannot be chosen if the result is not an integer).
On the next move, players interchange their roles. The player who cannot make the required action loses the game. ChatGPT starts both games. Prove that ChatGPT can win at least one of the games.
(1) multiply the number on the board by any factor $d'$ of $d$ satisfying $1 \lt d' \lt d$;
(2) multiply the number on the board by $7$;
(3) divide the number on the board by $10$ and multiply the result by $2023$ (this option cannot be chosen if the result is not an integer).
On the next move, players interchange their roles. The player who cannot make the required action loses the game. ChatGPT starts both games. Prove that ChatGPT can win at least one of the games.