Alright, let me put it simply like I’d explain it to a friend: You’ve got 100 lockers and 100 students. Each student changes the state (open/close) of lockers based on their number — like student #5 toggles every 5th locker. Now, here's the fun part: a locker ends up open only if it's toggled an odd number of times. And guess what? That only happens if the locker number has an odd number of divisors. And which numbers have an odd number of divisors? Perfect squares. Because divisors usually come in pairs like (2 × 6 = 12), but perfect squares have one unpaired divisor (like 4 × 4 = 16). So the lockers that stay open at the end are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 the perfect squares up to 100. 10 lockers in total stay open.
Alright, let me put it simply like I’d explain it to a friend:
You’ve got 100 lockers and 100 students. Each student changes the state (open/close) of lockers based on their number — like student #5 toggles every 5th locker.
Now, here's the fun part: a locker ends up open only if it's toggled an odd number of times. And guess what? That only happens if the locker number has an odd number of divisors.
And which numbers have an odd number of divisors? Perfect squares.
Because divisors usually come in pairs like (2 × 6 = 12), but perfect squares have one unpaired divisor (like 4 × 4 = 16).
So the lockers that stay open at the end are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 the perfect squares up to 100.
10 lockers in total stay open.