There are 100 lockers in a school hallway and they are all closed. 100 students come through the hallway and start opening and closing lockers. The first student opened all the lockers The second student closed every second locker The 2nd, 4th,6th and so on were closed The third student changes the state of every 3rd locker. In other words the student visits lockers 3 6 9 and if the locker is open it gets closed. If it is closed it gets opened The fourth student changes the state of every fourth locker, the fifth student changes the state of every fifth locker and so on until the 100th student changes the state of the 100th locker Which lockers are open after all 100 students pass through the hallway

Looking at this, we can see that each locker is only acted upon if the student number is a factor of it. This is why the 3rd person changes the state of every 3 lockers, the 4th student changes the state of every 4 lockers, etc.

The factors of a number are the numbers that the original number is evenly divisible by. When we divide these, we also receive a factor. With this information, we can conclude that unless one of the factors is the square root of the number, then the "second divided by" factor is itself. For every other number, there will be a "second divided by" factor, so an even number of them. Because an even number + 1 = odd number, then only perfect squares will have an odd number of factors.

Hope this helped!