There are 200 lockers in your school, numbered 1 thru 200 consecutively. The lockers are all closed to begin. Student #1 walks down the corridor and opens all the lockers that are numbered with a multiple of 1, thereby all the lockers are opened. Student #2 then walks down the corridor and changes the status (closed lockers are opened; open lockers are closed) of all the lockers that are numbered with a multiple of 2 (note that in the case of Student #2, this only involved closing lockers).
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What's the question?
Student #3 then walks down the corridor and changed the status of all the lockers that are numbered with a multiple of 3. By the end of this scenario, 200 students will have walked down the corridor, in numerical order, with each student changing the status of those lockers that are numbered with a number that is a multiple of the student's number.
At that point, which of the lockers are open? More importantly, why are these lockers open?