Locker Problem

For reference, I have seen and tackled this problem before and it is difficult for me to completely recall how I approached it the first time around.

I remember experimenting with the problem trying out a few "small" numbers first to see what would happen. I quickly realized that this was a factoring problem and looked at the factors of the numbers I tested. I believe I experimented all the way to 26 when I realized it was the perfect squares which were closed due to the property that they had an odd number of factors. Once I realized this the problem was "broken" and could be generalized for any number of students and lockers. The mathematician in me though, wanted to understand why this worked and so, I also wrote a small proof after solving it.