Question

Problem:
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.

Answer 1

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!)
Problem Hints:
Try playing sample games with different strategies (but always starting with 42000, with Yi making the first move). Try to figure out why the same player always wins the sample games.

Answer 2

Help? I need not only to have an answer, but explain it clearly with full sentences and correct grammar.

Answer 3

What..?

Answer 4

Sorry, there were just a lot of people who'd view it and then leave when I asked what they thought.

Answer 5

So..what do you think?

Answer 6

I think it's a tough problem.

Answer 7

It is, that's why I need help, LOL.

Answer 8

:P

Answer 9

I want to know, how do you win the game?

Answer 10

My family s never home; I haven't gotten a chance to play it.

Answer 11

...

Answer 12

...

Answer 13

Oh, I see.. nevermind

Answer 14

Sorry

Answer 15

So you start with the number 42000. And Yi gets to go first.

Answer 16

So,...Any ideas? Look at the problem hint. It's supposed to be the key to the solution

Answer 17

Prime Numbers = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and so on..

Answer 18

Yes..I knew that much.

Answer 19

Oh, I get what they're doing.

Answer 20

It says: "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."

This is called factoring.

Answer 21

OK...I think I'm following.

Answer 22

So, I think you will have to first find the factors of 42000

Answer 23

Here are all the factors of 42000:

(21000,2)
(14000,3)
(10500,4)
(8400,5)
(7000,6)
(6000,7)
(5250,8)
(4200,10)
(3500,12)
(3000,14)
(2800,15)
(2625,16)
(2100,20)
(2000,21)
(1750,24)
(1680,25)
(1500,28)
(1400,30)
(1200,35)
(1050,40)
(1000,42)
(875,48)
(840,50)
(750,56)
(700,60)
(600,70)
(560,75)
(525,80)
(500,84)
(420,100)
(400,105)
(375,112)
(350,120)
(336,125)
(300,140)
(280,150)
(250,168)
(240,175)
(210,200)

Answer 24

Remember they can ONLY divide 42000 by prime numbers, so we can get rid of the factors that aren't prime numbers.

Answer 25

OK....So, we first get rid of all the ones that are even, right?

Answer 26

@greenlegodude57 ?

Answer 27

(21000,2)
(14000,3)
(8400,5)
(6000,7)

These are the ONLY factors of 42000 that are PRIME.

Answer 28

OK.

Answer 29

Then you will have to find the factors of those factors..

Answer 30

The factors of 21000 that are prime are:

(10500,2)
(7000,3)
(4200,5)
(3000,7)

Answer 31

I get it now, you can only choose between 4 numbers when you divide this number.

Answer 32

OK.

Answer 33

Let's try an experiment:
Let's say Yi divides 42000 by 7:
42000
Sue is left with: 6000
Sue divides by 2:
Yi is left with: 3000
Yi divides by 3:
Sue is left with: 1000
Sue divides by 2:
Yi is left with 500
Yi divides by 5:
Sue is left with 50
Sue divides by 5:
Yi is left with 5:
Yi divides by 5
Sue is left with 1 and Yi wins.

Answer 34

No, if Sue divides 500 by 5 Sue is left with 100.

Answer 35

Oh right, my bad.

Answer 36

Let's say Yi divides 42000 by 7:
42000
Sue is left with: 6000
Sue divides by 2:
Yi is left with: 3000
Yi divides by 3:
Sue is left with: 1000
Sue divides by 2:
Yi is left with 500
Yi divides by 5:
Sue is left with 100
Sue divides by 2:
Yi is left with 50:
Yi divides by 5
Sue is left with: 5
Sue divides by 5
Yi is left with 1 and Sue wins.

Answer 37

It's fine.

Answer 38

OK....

Answer 39

Ah, I messed up again.

Answer 40

Let's say Yi divides 42000 by 7:
42000
Sue is left with: 6000
Sue divides by 2:
Yi is left with: 3000
Yi divides by 3:
Sue is left with: 1000
Sue divides by 2:
Yi is left with 500
Yi divides by 5:
Sue is left with 100
Sue divides by 2:
Yi is left with 50:
Yi divides by 5
Sue is left with: 10
Sue divides by 5
Yi is left with 2
Yi divides by 2
Sue is left with 1 and Yi wins.

Answer 41

OK.

Answer 42

So for most of the time they can only divide by 2 and 5. Because the result of the division will always have even numbers. And you can only sometimes divide even numbers with odd numbers, which is why we rarely used 7 and 3, only in the beginning.

Answer 43

The player is only left with 2 choices after a while, and they all lead the same way. Usually(when you're sue) if you pick 2 instead of 5 you're just making the game longer, because Yi will always win no matter what.

Answer 44

I hope that helps you, I have to go now.

Answer 45

@austinL should be able to help you.

Answer 46

OK...

Answer 47

prime factorization of 42000 :

\(42000 = 2^4 3^{1}5^37^1\)

Answer 48

that means, there are \(9\) prime factors in \(42000\)

Answer 49

so, the \(9th\) time who ever picks the result loses cuz, he gets to divide by the "last 9th prime factor" and gets a quotient of "1"

Answer 50

OK.

Answer 51

in the game, can u figure out who picks the result "\(9\) th " time ?

Answer 52

Yi or Sue ?

Answer 53

Well, since Yi goes first, Sue goes second, then Yi does third, then Sue goes fourth, then Yi goes fifth, then Sue goes sixth, then Yi goes seventh, which means Sue goes eight. SO Yi will lose.

Answer 54

? Is that right?

Answer 55

Correct !
so can we conclude this : whoever starts the game first will loose ?

Answer 56

So whoever goes first will lose, that's the solution? It's so simple and brilliant!

Answer 57

Thank you so so so much! You are a life saver.

Answer 58

^^ you got it ! u wlc :)