# Homework Help: 100 hotel rooms and 100 guests?

1. Jul 14, 2010

### mimibajiduh

there are 100 hotel rooms and 100 guests.
change the status means open goes to closed and closed goes to open.

guest 1: goes to every room and changes its status
guest 2: goes to every 2 rooms(2, 4,6,8,...,100) and changes its status
guest 3: goes to every 3 rooms and changes its status
guest 4: goes to every 4 rooms and changes its status
this goes on all the way to guest 100
guest 100: goes to 100 and changes its status

which rooms are closed? i know that room 1 is closed.

2. Jul 14, 2010

### tiny-tim

hi mimibajiduh!

take rooms 10 20 30 and 60 as examples …

how many guests visit each of those rooms?

3. Jul 16, 2010

### utkarsh1

I remember this question from my gr.7 class and I remember writing like huge essay to explain it. Lets see if i can do it in few lines. So, you have 100 rooms. or essentially 100 numbers. Then think about factors of each number. So lets say 8 for example factors are: 1, 2, 4, 8.
Now think about pairs: 1 x 8 = 8 and 2 x 4 = 8.
basically all of the numbers are pairs. Except 10 of them; namely squares: 1,4,9,16,25,36,49,64,81,100..... By that i mean consider factors for 36 =>
1 x 36 = 36
2 x 18 = 36
3 x 12 = 36
4 x 9 = 36
BUT:
6 x 6 = 36

Hopefully you get the idea.

4. Jul 22, 2010

### utkarsh009

hey utkarsh 1 , i am utkarsh009, what a coincidence!!!!
first tell me what was the original status of all the rooms and also whether the guests have to change the status of their room for coming out?
now, let me just explain what you said in my words. just see if i am correct or not.
the the numbers which have even number of factors have the same status and the ones with odd number of factors have same status.
if the original status was that all the rooms were opened then the square numbers (which have odd number of factors) will be closed.
just check whether i am right or not?

5. Jul 23, 2010

### utkarsh1

Yes its correct! Probably better explained since its more concise

6. Jul 23, 2010

### utkarsh009

thank you very much !!!!!!!!!!!!!