A question to test ur "MATH BRAIN"

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ryan_kidz

Junior Member
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Sep 11, 2005
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The Question is:
In a strange school, there'r..

1000 lockers + 1000 Students
The first student opened all the doors of the lockers
The second student closed all the doors of the lockers #2,4,6,8,10...
The third student reversed the doors of #3,6,9,12,15...(ie If the door was closed, he 0pened it and if the door was opened he closed it)
and the 4th student reversed the doors of #4,8,12,16,20....
....
....
After all these 1000 students, which doors remained closed?

JUSTIFY UR ANSWER!
 
Google the problem. It's been done to death, and there are loads of solutions posted online that you can print out and hand in.

Eliz.
 
The Lockers Paroblem

We could take the long route and discuss the evolution of the answer but lets try a shorter path.

Lets look at locker number "n". All the lockers are locked to begin with. Since all the lockers are opened on the 1st pass, locker "n" is now open. For locker "n" to be closed on the 2nd pass, n must be divisible by 2. For the locker to be opened on the 3rd pass, it must be divisible by 3. For the locker to be closed on the 4th pass, it must be divisible by 4. Clearly, the locker is either opened or closed as long as the locker number is divisible by each successive divisor of "n". After the one thousandth student has made his contribution to the celebration, locker "n" will only be open if it was acted upon an odd number of times. We can therefore conclude that locker "n" will be open if, and only if, the number "n" has an odd number of factors or divisors. But the only numbers that have an odd number of factors/divisors are the perfect squares. Thus, if "n" is open, it is one of the perfect squares. The lockers that remain open are therefore numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, on up to 961.

For each locker number, find all of the exact divisors including 1 and the number itself. If the number of divisors is odd, then the number of people who reversed the locker is odd, and the locker is open. If the number of divisors is even, then the number of people who reversed the locker is even, and it is closed. The only numbers with an odd number of divisors are the perfect squares. Therefore, the lockers that remain open are those identified by the perfect squares.
 
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