another brain teaser for the math inclined

[lotus776]

1000 lockers at WENUMBEREM HS are displayed in one long row and are labeled from 1 to 1000. In addition, 1000 students are lined up in a row at locker number 1. The first student runs and opens every locker. The third student runs through and CHANGES every third locker; that is, if it is open, he closes it; if it is closed, he opens it. The fourth student runs through and CHANGES every fourth locker. This process continues until the 1000th student runs through and changes the 1000th locker.
1) At the end of this process, which numbered lockers are left open?

2) What is the pattern of numbers in (1)? BE VERY SPECIFIC!

3) what is the reasoning for this pattern? EXPLAIN IN DETAIL!

good luck