Which lockers are open after 1000 students toggle them

[Alethia]

1. A new high school has just been completed. There are 1,000 lockers in the long hall of the school and they have been numbered from 1 to 1,000. During lunch, the 1,000 students decide to try an experiment.

- The first student, student 1, runs down the row of lockers and opens every door.
- Student 2 closes the doors of lockers 2, 4, 6, 8 and so on to the end of the line.
- Student 3 changes the state of the doors of lockers 3, 6, 9, 12 and so on to the end of the line. (the student opens the door if it is closed and closes the door if it is opened)
- Studnet 4 changes the state of the doors 4, 8, 12, 16 and so on. Student 5 changes the state of every fifth door, student 6 changes the state of every sixth, and so on until all 1000 students have had a turn.

When the students are finished, which lockers doors are open?
---
I was REALLY tempted to sit tehr are write out numbers 1-1000 but decided there was an easier way...

 

[KLscilevothma]

which lockers doors are open?

Try the case when there are only 10 lockers and try to find out a pattern. Then try to explain why the pattern is like this.

 

[Njorl]

There won't be a pattern. Take the factors of each number, include 1 and the number itself. Count the number of factors. This is the number of "state" flips. If it is even, the locker is closed. If it is odd, the locker is open.

Njorl

 

[Hurkyl]

There most certainly is a pattern.

 

[Njorl]

Sorry,
I had to do a few more lockers. You're right Hurkyl. For anyone else trying this, more power two ya'.

Njorl

 

[jimbot]

One certainty

Prime numbered lockers will be closed.

 

[Njorl]

There's more two it than that. Many other lockers will be closed too.

Njorl

 

[pappi.51123]

All the square numbers will be open...