Solve High School Lockers Calculus Problem

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The problem involves 1000 lockers and students toggling the doors based on divisibility by their locker numbers. Each locker starts closed, and students change the state of lockers that divide their own number. A suggested approach is to simulate the first 15-20 students to identify a pattern in the toggling process. The solution hinges on understanding how many times each locker is toggled, which relates to the number of divisors it has. Ultimately, only lockers that are perfect squares will remain open at the end.
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Hi everyone, i recently got this problem from my college professor and I am either incredibly rusty or just don't know how to do this. I've tried to set up some sort of equation using f(x) but i just can't make anything fit or account for everything. any help would be appreciated.

the question follows:
A high school has 1000 students and each has a numbered locker where they keep various smelly items. Fortunately all the locker doors are shut. One by one, each student walks past the lockers, and either opens or shuts (depending on its previous position) the door of any locker that divides their own locker number. How many lockers are open at the end?
 
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My daughter got this same problem a few weeks back. Suggest you just plug-n-chug to map out the 1st 15-20 students until you see the pattern. At least, that's how we did it.
 

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