Answer: Option A
:
A
Let us take some simple numbers to see if there is a pattern
If you take a number, say 6, In the first round it will be on, 2${}^{nd}$it will be off, 3${}^{rd}$ it will be on, and 6${}^{th}$ it will be off. Hence its final state will be off.
Consider a number 4-In the first, second and fourth round it will be ON, OFF & finally ON
Consider a number 9. in the first, third and 9${}^{th}$ round it will change its state and its final state will be on
From this, you can deduce that the state of the bulb is determined by the property of the number; i.e. all numbers with odd number of factors will remain in an "on” state and all other numbers will be off. Conclusion- All the bulbs which are on are "perfect square” numbered bulbs
Thus, the bulbs which will be on are all in the positions of numbers which are perfect squares.
The question is thus, asking for the number of perfect squares till 1000= 31 (i.e. 1${}^2$,2${}^2$...........31${}^2$)