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

Here's how Khoisnam figured out which lockers would be open: Step 1: Understanding the Locker Toggling 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. Step 2: Determining How Many Times Each Locker is Toggled 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. Step 3: Identifying Lockers That Remain Open 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. Step 4: Listing the Perfect Squares Khoisnam would have identified all the perfect squares between 1 and 100. These are the numbers of the lockers that will remain open: 1^2 = 1 2^2 = 4 3^2 = 9 4^2 = 16 5^2 = 25 6^2 = 36 7^2 = 49 8^2 = 64 9^2 = 81 10^2 = 100 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: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100