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?

Answer 1

The first person starting will always lose.

Explanation:

First, I will the prime factorization of 42000. After finding the prime factorization of 42000, I will count the number of primes in the prime factorization. After that, we will find which round the person will lose.

Prime Factorization of 42000: 2*2*2*2*3*5*5*5*7

Number of Primes in List: 9

The person losing: The person on the 9th turn will lose.

The person on the 9th turn is Yi. Therefore, no matter how Yi or Sue keeps mixing the order of the numbers, Yi will always lose.

I hope this helps!

Answer 2

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.

Yi is going to lose.

Have an awesome day! :)