Math Problem Involving 1000 doors

[mck3939]

There are 1000 doors. Each one labeled with a number 1-1000. A person opens all doors whose number has one as a factor (which is all of them). Then she closes all doors whose number has two as a factor. Then the person continues to change the status of the doors (opening or closing them) based on the number and factors. What lockers will be open when we reach 1000? How do you show your work?

 

[ineedhelpnow]

Hi. Welcome to MHB. (Wave) We ask you to post any work you have tried or any effort and to explain where you are stuck or having trouble therefor our math helpers or anyone who can help has a better idea and understanding of what you have tried so far and where you are standing/stuck. Thank You. :)

 

[mck3939]

There are 1000 doors. Each one labeled with a number 1-1000. A person opens all doors whose number has one as a factor (which is all of them). Then she closes all doors whose number has two as a factor. Then the person continues to change the status of the doors (opening or closing them) based on the number and factors. What lockers will be open when we reach 1000? How do you show your work?

I tried first dividing 1000 by 2 to get 500
then 1000 by 3 to get approximately 333
and 1000 by 4 and so on, but I find this is taking forever and is no the best strategy to use. I cannot think of a better one.

 

[I like Serena]

There are 1000 doors. Each one labeled with a number 1-1000. A person opens all doors whose number has one as a factor (which is all of them). Then she closes all doors whose number has two as a factor. Then the person continues to change the status of the doors (opening or closing them) based on the number and factors. What lockers will be open when we reach 1000? How do you show your work?

I tried first dividing 1000 by 2 to get 500
then 1000 by 3 to get approximately 333
and 1000 by 4 and so on, but I find this is taking forever and is no the best strategy to use. I cannot think of a better one.

Hi mck3939! Welcome to MHB! :)

Let's start with door 1.
We open it... and we're done, since 1 is the only number that divides 1.

Next is door 2, which is a prime.
We open it, we close it, and we're done.
So we manipulate it twice, since 1 and 2 are the only numbers that divide 2.

How about, say, doors 3 to 10? (Wondering)

 

[CellCoree]

This problem can be solved using the concept of prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves. The only prime numbers between 1 and 1000 are 2, 3, 5, and 7.

First, we know that all doors will be open after the first step, since all numbers have 1 as a factor.

In the second step, the person will close all doors whose numbers have 2 as a factor. This means that doors with even numbers will be closed, leaving only the doors with odd numbers open.

In the third step, the person will close all doors whose numbers have 3 as a factor. This will close doors labeled with multiples of 3, leaving only the prime numbers (2, 3, 5, and 7) open.

In the fourth step, the person will close all doors whose numbers have 5 as a factor. This will close doors labeled with multiples of 5, leaving only the prime numbers (2, 3, 5, and 7) open.

In the fifth step, the person will close all doors whose numbers have 7 as a factor. This will close doors labeled with multiples of 7, leaving only the prime numbers (2, 3, 5, and 7) open.

At this point, all doors with prime numbers as their labels will be open, and all other doors will be closed. This means that the only open doors when we reach 1000 will be doors labeled with 2, 3, 5, or 7.

To show this work, we can create a table or chart with the numbers 1-1000 and mark which doors are open or closed after each step. This will clearly show the pattern of closing doors with certain factors and leaving only doors with prime numbers open.