Math from the past

The kids were all great at math. I tried to assist Andrew only once. Here is the one problem. I think he was in Jr High at the time, grade 7 or 8.

The School Locker Problem

Once there was a school that had exactly 1000 students. The student population had been exactly 1000 for many years, never varying, so the school had built exactly 1000 lockers – one each for the 1000 students. Each of the students was assigned a locker each year. The lockers were numbered from 1 to 1000. At the start of the year, the students performed a ritual, “Preparing the Locker Doors”. This is the story of that ritual.

To start, all the locker doors, all 1000 of them, were closed.

The students lined up according to their locker number in ascending sequence from 1 to 1000.

The first student, Locker 1, went down the whole row of 1000 lockers and, starting at Locker 1, opened every locker door.

The second student, Locker 2, started at the second locker and closed every second locker door.

The third student, Locker 3, started at the third locker and closed the door, then proceeded to either open or close every third locker as required.

And so it continued with each student starting at their locker “N” and proceeding to the next “Nth” locker, either opening or closing the locker door as required. If the locker door was closed when they arrived, they opened it. If the locker door was open when they arrived, they closed it. But they looked at only every “Nth” locker door.

This process continued until all 1000 students had performed the task.

At the end of the process,

how many doors were open, how many were closed, and why?