Open lockers

A school has a hallway with 100 lockers. All the lockers are closed.

A student walks down the hallway can opens every locker. A second student walks down the hallways and closes every second locker, starting at locker 2. A third student then walks down the hallways and changes the state of every third locker, starting at locker 3.

The $n$th student walking down the hallway changes the state of every $n$th locker starting at locker $n$.

After 100 students have passed down the hallway, which lockers are open?

The lockers left open are the square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

Proof

The lockers that are left open are those that have been changed an odd number of times.

Since student $n$ changes all the lockers that have $n$ as a factor, the lockers that have changed are those that have an odd number of factors.

Factors come in pairs ($n=f_1\times f_2$), so for there to be an odd number of factors, a factor $f$ must repeat itself in a pair ($f=f_1=f_2$). In this case $n=f\times f$, thus $n$ is a square number.