The factors of 81 are 1, 3, 9, 27, and 81.
Bulb 81 is turned on initially. Then it's turned off by the 3's flip, on by the 9's flip, off by the 27's flip, and on by the 81's flip. That's because 9 * 9 = 81, so there is no corresponding factor to 9 to cancel out its flip.
*Edit*: Art T's explanation of primes is interesting, but in this case useless because it doesn't apply to this problem. That can be demonstrated very easily, as he claims that the prime-numbered bulbs should remain on, which include bulbs 2 and 3, while the composite-numbered bulbs, like 4 should be turned off. Let's examine that. All bulbs start "on", then someone comes out and turns off all the multiples of 2. So bulbs 2 and 4 are turned off. We immediately see that there's a problem, because the next person flips the multiples of 3, so no one ever turns bulb 2 back on. Bulb 3 gets turned off by that next person, and no one ever turns it back on, so that's two primes in a row that remain off. Bulb 4, which was turned off by the multiples-of-2 person, gets turned back on by the multiples-of-4 person, and never gets turned back off, so it remains on. We now have two primes that remain off, and a composite (which happens to be a square) that remains on. All three of those contradict the assertion in Art T's response. See my explanation above of why it's the squares that remain lit, rather than the primes.