Challenge Problem that seems too Easy

[brtdud7]

This is a challenge problem I received in my calculus class that I believe uses math lower than Calculus. My worry is that the way I want to solve it seems too simple.

Homework Statement

A long hallway contains lockers numbered 1 through 1000. At the start of the day, all of the lockers are closed. Someone comes by and opens every other locker, starting with locker #2. Then someone else comes by and changes the "state" (ie, closes open lockers and opens closed lockers) of every third locker, starting with locker #3. Then someone comes by and changes the state of every fourth locker starting with #4, and so on, until no more lockers can be changed in this way. How many lockers are closed at the end of this process?

The Attempt at a Solution

I was just going to make a table, first with #2 to #1000 (by 2s) all open, then make one with all the multiples of 3 with each locker's state, then make one with all the multiples of 4 and the resulting state. It just seems too algorithmic for a calculus class, so I was curious if this brute force method is correct, or if anyone is aware of any formulas or theorems that could be used instead to make this problem less time-consuming?

 

[icystrike]

Think logically. Start with the range from 1 to 10.. Then to hundred...

 

[Avodyne]

You're right that this has nothing to do with calculus. Here are some questions to ask yourself.

Given a number N, how many times is the state of the Nth locker changed? (Hint: the answer has to do with the factors of N).

Under what conditions is the number of state changes of the Nth locker an even number?

 

[brtdud7]

Oh alright, I figured that it had something to do with the LCM of the lockers, like the LCM of 2 and 3 is 6, so every 6th locker's state would change. I was going to start with that and then see where I go from there.