Q1.
Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer? Hint: Find out how many times each locker is toggled.
Answer
He counted factors. Person k touches locker n exactly when k divides n, so locker n is toggled once for every factor of n.
Locker 6 is touched by persons 1, 2, 3, 6 → 4 toggles
Every locker starts closed, so it ends open only after an odd number of toggles
Every locker starts closed, so it ends open only after an odd number of toggles
So the question becomes: which numbers from 1 to 100 have an odd number of factors?
Why it happens: Factors come in partner pairs — if d divides n, then so does n ÷ d, and the two multiply to n. Pairing off every factor with its partner would make the count even. The count can only be odd when one factor is its own partner, that is when d × d = n. That happens exactly for the squares.
So Khoisnam knew the answer before anyone moved: the lockers left open are
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Check it yourself: Locker 16 is touched by 1, 2, 4, 8, 16 — five people, an odd number, so it stays open. Locker 12 is touched by 1, 2, 3, 4, 6, 12 — six people, so it closes.