1000 Prisoners in a cell
There are 1000 cells in a prison.Each cell has a prisoner. When i divide all the cell by a number (starting with 1 to 1000) and if the remainder is 0, then the door will be inverted (i.e. if it's close then it will open or vice versa). When i keep those cells on dividing till 1000, at last how many cells will be opened? Initially all doors are closed..... Let's take an example:- if i divide all the cells(1 to 1000) by 1,then remainder will be 0 for all cells. Therefore every cell will open.Now, if i divide again by 2 then only even number's cell will close.I continue this process till 1000...How many cells are open????
The answer is 31.
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A cell will remain open if it has been switched odd number of times.
Any number (prime or not) can be written as product of 2 different numbers.
Let's take example of 12: 12=1x12 =2x6 =3x4
So 12 will switch 6 times on: 1, 2, 3, 4, 6, 12
But we can notice that square numbers will switch odd times. Say
36=1x36 =2x18 =3x12 =4x9 =6x6
So 36 will switch 9 times on: 1, 2, 3, 4, 6, 9, 12, 18, 36
Between 1&1000, 31 squares are there (31²=961)
So answer=31