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Opened And Closed Lockers
A school hallway has a thousand lockers. A student goes down the hallway opening every locker. Then a second student goes down the hallway closing every other locker. A third student goes down the hallway, and at every third locker, he opens it if it was closed and closes it if it was opened. This continues on for a total of 1000 students opening and closing lockers. How many lockers are opened when the students are finished?
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8 Answers
62This one uses a brilliant property of numbers.. I dont know how many of u realised what property to check..
basically u need to look at the number of ******** of a number! (That will give u the solution!)
62yes varun.. brilliant work
now find which ones have the property u are talking about?
1Is the answer 31 ? i.e 31 lockers are open ?
All the squares :P ( < 1000 )
1it is based on the principle that perfect squares have odd number of factors and non squares have an odd number of factors.here is my proof.
let x be a non square number. if 'a' is a factor of x, another factor will be 'x/a'. so its factors exist in pairs
if p^2 is a perfect square, if 'b' is a factor of p^2, another factor will be (p^2)/b. but here there is a special case. when 'b'='p' so p and (p^2)/p are equal. so all factors of p^2 can be grouped as pairs except p so a perfect square has odd number of factors.
