The students at Prime High School have a tradition for spirit days. The first st

Question

The students at Prime High School have a tradition for spirit days. The first student opens each of the 1000 lockers in the school. Then the second student goes through and closes every second locker. The third student changes the state of every third locker. That is, if the locker is open, she closes it. If the locker is closed, she opens it. The fourth student changes the state of every fourth locker. This continues with every student until the 1000th student comes in and changes the state of the 1000th locker. After all students have gone through, which lockers are open? Why?

Explanation / Answer

so locker is open if it has been worked upon odd number of time

while the locker is closed if it has been worked upon even number of time.

1. Prime numbers

they will be closed because they will be worked upon twice

2. Perfect square

they will be opened because they will be worked upon odd number of times.

e.g. 4, 16

these numbers have odd number of factors

3. Non perfect square

they will be closed because they will be worked upon even number of times

e.g. 5, 8, 10, 15  

3. 1

1 will be opened.

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