The Locker Problem

Problem

Imagine a long hallway containing 100 lockers, all initially closed. One hundred students walk down the hallway one at a time and each follows a simple rule:

• Student 1 toggles (opens if closed, closes if open) every locker: 1, 2, 3, ..., 100.
• Student 2 toggles every 2nd locker: 2, 4, 6, ..., 100.
• Student 3 toggles every 3rd locker: 3, 6, 9, ..., 99.
• In general, Student k toggles every k-th locker.

After all 100 students have walked the hallway, which lockers are left open? How many are there?