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?
Accepted Solution
A:
The prime factorization of 42,000 is 2×2×2×2×3×5×5×5×7.
There are a total of 9 prime number factors of 42000.
That means that the quotient can only be passed 9 times until someone reaches 1.
The person on the 9th turn will lose.
In this case, the person who has the 9th turn is Yi.