Locker Problem

Super scratchy work on the problem shown below. I was basically thinking out loud on the page. The first question I asked myself was "how many times will a given locker have its state changed?" I realized that it would be determined by the number of factors of the locker's number, and went from there. You'll notice that I started with student 1 closing all of the lockers, and then basically worked on the problem starting with student number 2 and all lockers being closed. I get a little hung up on excluding 1 until near the end. I began starting to work out how to find the number of factors a given number would have, based on prime factorizations. I'm thinking it would involve counting the number of combinations of prime factors, but I realize it falls apart because it double counts factors.

At this point I've finally started to realize how to count the number of factors nicely based on the prime factorization, and I also keep on excluding 1 for some reason (I think it's just how the problem was pitched in class that got me stuck thinking like that).

I have a nice formula for giving me the number of distinct factors, and finally decide to include 1 because I realize the formula automatically does that. I swap around even and odd in my "final answer" to account for this change.

Ran through an example number just to try it out. Realized I would need to run every number through the formula to get a complete list and decided to call it quits. I should have known from the way the question was phrased that the answer would turn out to be an easily described set of numbers, but it honestly never occurred to me that only perfect squares have an odd number of factors - cool problem!