suppose you are in a hallway lined with hundred closed lockers.you begin by opening every locker. then you close every second locker.then go to every third Locker and open it( if its close) or closed if (it is open) .let's call this action toggling every ninth locker on pass number n. after 100 passes, where you toggle only locker #100,how many lockers are opened ?
Answers
Answer:
Ten lockers are left open after 100 passes.
Step-by-step explanation:
Initial position of all the 100 lockers is 'Close'.
For the first locker, it is toggled only once, so, it will be kept open.
For the second locker, it is toggled at the pass 1 and 2. So, it is left closed.
For the third locker, it is toggled at the pass 1 and 3. So, it is left closed.
For the fourth locker, it is toggled at the pass 1, 2 and 4. So, it is left open.
For the fifth locker, it is toggled at the pass 1 and 5. So, it is left closed.
…..
For the ninth locker, it is toggled at the pass 1, 3 and 9. So, it is left open.
Similarly, for the 100th locker, it is toggled nine (odd) times (i.e,) at the pass 1, 2, 4, 5, 10, 20, 25, 50 and 100. So, it left open .
It can be noticed that if the locker is toggled for even number of times, then it is left closed.
And, for the odd number of toggles, it is left open.
We know that only the perfect square (numbers) have odd number of factors. Eg. Factors of 4 are 1, 2 and 4.
So, only the ten locker numbers (perfect squares) such as 1, 4, 9,16,25,36,49,64,81 and 100 will be left open.
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