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?

Question
Answer:
The prime factorization of 42,000 is 2Γ—2Γ—2Γ—2Γ—3Γ—5Γ—5Γ—5Γ—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.

Yi is going to lose.

Have an awesome day! :)
solved
general 11 months ago 3184