The Corridor of 500 Lockers

A school corridor contains $500$ lockers numbered $1$ through $500$, and every door starts open. Then $500$ students file through one at a time: student $n$ (for $n = 1, 2, \dots, 500$) flips the state of every locker whose number is a multiple of $n$ — closing it if open, opening it if closed. So the first student flips every locker, the second flips lockers $2, 4, 6, \dots$, and so on through student $500$. Once everyone has passed, how many lockers stand open?