Locker Problem

Problem
There is a school with $100$ students, and correspondingly $100$ lockers, all of which start off closed.

The first student opens every locker.

The second student closes every other locker, starting with the second.

The third student changes the state of every third locker, starting with the third.

That is, if the locker is open, he or she closes it, and if it is closed, he or she opens it.

This continues similarly until all $100$ students have passed along the lockers.

After the $100$th student has finished, which lockers are open and which are closed?

Solution
The square numbered lockers ($1, 4, 9, \ldots$) are open, and all others are closed.