To solve this problem, we need to analyze how many times each bulb is toggled and determine which bulbs end up being on.

Key Observations:

1. Toggle Rule: A bulb starts off. It is on if toggled an odd number of times (since even toggles return it to off).
2. When is a bulb toggled?: Bulb n is toggled by person k if k is a divr of n (person k toggles every k-th bulb).
3. Divrs: Divrs of a number come in pairs (e.g., 6 → 1&6, 2&3). Only perfect squares have an odd number of divrs (their square root is a repeated divr: e.g., 4 →1,2,4).

Count Perfect Squares:

We need the number of perfect squares ≤1000.

• $31^2 = 961$ (≤1000)
• $32^2 =1024$ (>1000)

Thus, there are 31 perfect squares from $1^2$ to $31^2$.

Answer: 31

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