A shortcut to finding the answer is explained here:
http://www.daniweb.com/blogs/entry387.html#
Only locker numbers that are perfect squares will be left open after all 500 students have done their thing. The series will be the perfect squares from 1, 4, 9 etc. up to 22^2 = 484.
Newon Middle School has a hall with 500 lockers, all of which are closed. Five hundred students start down the hall. The first student opens every locker. The second student closes all the lockers that are multiples of 2. The third student changes (closes an open locker or opens a closed locker) all multiples of 3. The fourth students changes all multiples of 4. And so on. After all students have finished with the lockers, how many will be closed and how many will be open?
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