I'm thinking about a game theory problem related to factorization. Here it is,
Q: two players A and B are playing this factorization game. At very first, we have a natural number $270000=2^4\times 3^3\times 5^4$ on a chalkboard.
in their own turn, each player chooses one number $N$ from the chalkboard and erase it, and then write down new two natural numbers $X$ and $Y$ satisfying
(1) $N=X\times Y$ (2) $gcd(X,Y)>1$ (So, they are "NOT" coprime)
a player loses if he cannot do this process in his turn.
So, in this game, possible states at $k$-th turn are actually sequence of natural numbers $a_1,a_2,\dots,a_k$ with $a_i>1$ and $a_1\times a_2\dots \times a_k=270000$
EXAMPLE of this game)
So B loses in above case.
Actually, in this game, the first player(So, A) has winning strategy.
What is this winning strategy for A?
-Attempted approach: I tried to find what is "winning position" and "losing position" for this combinatorial game. but classifying these positions were not so obvious.
What is A's winning strategy? Thanks for any help in advance.