The Locker Problem

The general strategy I use for problem solving is to brute force a few numbers and look for patterns. When using brute force for the first 6 numbers, I figured that the number of students who changed the nth student's locker is the number of factors of n. Since the lockers are initially all open, I figured that an odd number of factors result in a closed locker and an even number of factors result in an open locker. That got me thinking: when specifically does a number have an odd number of factors? I figured that it only happens when the number is a perfect square (because factors pair up & a factor pairs up with itself only if the number is a perfect square). Thus, I concluded that the closed lockers are the ones that are perfect squares, and the open ones are the lockers that are not perfect squares.