In a school with 100 lockers, all initially closed, 100 students walk by. Student 1 toggles every locker. Student 2 toggles every second locker. Student 3 toggles every third locker, and so on, until

Each locker starts in a closed state. When a locker is "toggled," its state changes: if it's open, it closes; if it's closed, it opens.

• If a locker is toggled an odd number of times, it will end up open.
• If a locker is toggled an even number of times, it will end up closed.

Locker number $n$ is toggled by person number $k$ if $n$ is a multiple of $k$. This means that $k$ is a divisor of $n$.
Therefore, the total number of times a locker $n$ is toggled is equal to the number of its divisors.

For a locker to remain open, it must be toggled an odd number of times. This means its number must have an odd number of divisors.
A positive integer has an odd number of divisors if and only if it is a perfect square.

Khoisnam would have identified all the perfect squares between 1 and 100. These are the numbers of the lockers that will remain open:

Khoisnam figured out that the lockers corresponding to these perfect square numbers would be the ones to remain open.

The lockers that will be open at the end are:
$\text{1, 4, 9, 16, 25, 36, 49, 64, 81, 100}$