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brtdud7
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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.
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?
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?
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?