The answer to that was indeed that all the square lockers were open, because they had an odd number of factors, but this one will have a different answer.

If nothing else, the lockers numbered above 500 will have a pattern of open, close, something.

]]>Wouldn't all the prime numbered doors be shut, and all those with and even number of factors like #2 ( 1 & 2) the door would be closed, and those with an odd number of factors like locker number 9 (1 & 3 & 9) would be open? And the prime numbers would all have an even number of factors (1 & #), so the door would be closed?]]>

n = 5k

n = 5k + 1

n = 5k + 2

n = 5k + 3

n = 5k + 4

Note this covers all possible values for n. Now lets go through them one by one:

n = 5k. Then n^2 = 25k^2, which has a remainder of 0 when divided by 5.

n = 5k + 1. Then n^2 = 25k^2 + 10k + 1 = 5(5k^2 + 2) + 1, which has a remainder of 1 when divided by 5.

Go through the others.

]]>2) A school has 1000 students, and their lockers, which are numbered from 1 to 1000, are all closed, The first student opens all the lockers. The second student closes every second locker, beginning with her locker #2. The third student CHANGES the state of every third locker, beginning with locker #3, which means a locked locker becomes open and an open locker becomes closed. This carries on until all 1000 students had their turn.

Which lockers are open and why?

After some work, I think the locker with perfect square numbers are open. So locker #1, 4, 9, 16, 25 etc are open. But I dont know how to prove this and it might not work for numbers within 900-1000.

Also, I need help solving the first problem in this thread :

http://www.mathsisfun.com/forum/viewtopic.php?id=2859

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