Fun Math problem (More Light Bulbs)?

Question

If there are 20,000 light bulbs in a row, and they are all OFF.
All of the lights have one of them strings to turn it from on to off, or off to on..
First go down the row and pull every lights string.
So you pull lights 1,2,3,4,5...
Now do it again pulling every 2nd lights string
So you pull lights 2,4,6,8,10...
Now the 3rd, 3,6,9,12,15..
4th, 4,8,12,16...
Keep doing this until you get to the end. Then end is when you pull every 20,000th lights string...(which means you only pull the string of the last light)

At the end, how many lights are ON?
And explain which ones will be on...

Have fun...

Answer 1

The ones which have an odd number of distinct factors (including 1 and itself) will be ON.
The ones with an even number of distinct factors will be OFF.
(Interestingly all prime numbers will be OFF.)

Perfect squares are the only integers that have an odd number of distinct factors.

For every integer which is not a perfect square, its factors can be grouped pairwise
eg 30 = 1*30 = 2*15 = 3*10 = 5*6 (total 8)

For perfect squares, one of those pairs will consist of hte same factor twice, that's why perfect squares have an odd number of distinct factors.
eg 36 = 1*36 = 2*18 = 3*12 = 4*9 = 6*6 (total 9)

So the answer is the total number of perfect squares from 1 to N. sqrt(20000) = 141.4 which means there are 141 perfect squares from 1 to 20000.

ANS 141 ON
They are the bulbs in position 1,4,9,16,...19881

Answer 2

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