Estonian Mathematical Olympiad

As $2024!=2023!\cdot 2024$ and $2024=2^3\cdot 11^1\cdot 23^1$, the sum of the exponents of all primes in the prime factorisation of the number on the board is odd in one of the games and even in the other. We will show that ChatGPT can win the game where the sum is even.

On any of its moves, ChatGPT chooses $d$ to be a product of exactly two (not necessarily distinct) primes dividing the number on the board. Then, no matter what its opponent does, the sum of exponents will increase by $1$. This is obvious for the first two choices; for the third choice we notice that $10=2\cdot 5$ has two prime factors and $2023=7\cdot 17^2$ has three prime factors. Thus the move as a whole decreases the sum of exponents by $1$.

Whatever the opponent chooses as $d$, ChatGPT chooses the first option with

d'$=\frac{d}{p}$, where $p$ is any prime factor of $d$. Thus the move as a whole decreases the sum of exponents by $1$.

Continuing with this strategy, ChatGPT ensures that at the beginning of each of its turns the sum of exponents is even. As the sum of exponents decreases by one during every move, there will eventually be a situation where ChatGPT's opponent will start its move with the sum of exponents being $1$, i.e. with a prime on the board. Thus the player cannot make their move, meaning that ChatGPT wins.