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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?

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.

@bobpursley

so you mean we have to do a factor tree for 1000 numbers?!

Yeah, that's the tkciet, sir or ma'am