Problem of the Week #2

At a new junior high school, there are exactly 1000 students and 1000 lockers. Lockers are numbered in order from 1 to 1000. On April Fool’s Day the students played the following prank. The first student to enter the building opened every locker. The second student closed every locker that had an even number. The third student changed every third locker, closing those that were open and opening those that were closed. The fourth student changed every fourth locker, and so on. After all 1000 students passed through the locker room, which lockers were open?