Math Whizzes: I need help with a problem

[dave127]

I have this problem for my algebra II class and I dont know how to figure it out. I'm not so much interested in the answer as much as how you get it. Herest teh Prob":

There are 1000 lockers lined up, all closed, and a line of 1000 people. Suppose the first person goes along and opens every locker. The second person goes along and closes every other locker starting with the second locker. The third person goes along and changes the stateof every third locker (if it si open he closes it, and vise versa). The fourth person goes along and changes the state of every fourth locker, and so on until all the people have passed all the lockers. Which lockers are open in the end? Which locker is the last one open?

this is it, if anyone knows a formula or a way to figure it out wihtout drawing it out, let me know. someone said that you can enter a formula or something into a spreadsheet to do this, but i dont knkow what the formula is. if anyone knows please help

thanks
dave

 

[Bling Bling]

the last one open is the first locker. think about it: no one but the first person touches it, so it is never closed.

 

[dave127]

thanks bling bling, that sounds like a logincal answer. does anyone else have something to say, or is bling bling's answer right

 

[rhinox99]

your problem is basically this one:
http://forums.anandtech.com/messageview.cfm?catid=38&threadid=276327

 

[Napalm381]

I (under Toolgirls name) posted this in the other thread- here it is with some additions:

The perfect squares will be open, the rest will be closed. For all non-perfect squares, there are an even number of factors (i.e. 24- 1,2,3,4,6,8,12,24)- there must be an even number of factors, as they're all pairs. Each locker will be modified an even number of times, which means it will end up closed.

However, with the perfect squares, one "pair" of factors is a single number- (i.e. 5 and 5 for 25). So perfect squares are the only ones that can have an odd number of factors. They will be modified an odd number of times, which mean they will end up open.

The last perfect square under 1000 is 961- 31 squared.

 

[dave127]

thanks nalpalm, i understand this now. how did you figure it out or did it just dawn on you?