Question

Yi and Sue play a game. They start with the number $42000$. Yi divides by a prime number, then passes the quotient to Sue. Then Sue divides this quotient by a prime number and passes the result back to Yi, and they continue taking turns in this way.
For example, Yi could start by dividing $42000$ by $3$. In this case, he would pass Sue the number $14000$. Then Sue could divide by $7$ and pass Yi the number $2000$, and so on.

The players are not allowed to produce a quotient that isn't an integer. Eventually, someone is forced to produce a quotient of $1$, and that player loses. For example, if a player receives the number $3$, then the only prime number (s)he can possibly divide by is $3$, and this forces that player to lose.

Who must win this game, and why? Explain your reasoning in complete sentences. (Don't just show one example of how the game could go. Instead, explain why the same player must always win, no matter what strategy either player tries to use!)

Answer 1

42000 = 2^4*3*5^3*7, so has 9 prime factors

The player who plays 9th will lose. If only 2 people are playing, that is the person who takes the first turn. The second player in a 2-person game with this starting number will always win.