A high school has a strange principal. On the first day, he has his st

Question

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
 
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

A 45
B 31
C 77
D 131
E 93

Bunuel · Accepted Answer

Initially all lockers are closed. Now, consider one particular locker for example locker 6.

First student will open it. Then it will be closed when second student will close every second locker as 6 is a multiple of 2, then it will be opened again when 3rd student do the same as 6 is a multiple of 3 and finally it'll be closed when 6th student get to it as 6 is a multiple of 6.

1(o)-2(c)-3(o)-6(c) --> so # of variations of a locker open/close equals to # of factors of a locker and if the # of factors of a locker is even (as for 6) then it will end up locked and if the # of factors of a locker is odd it'll end up opened. Only perfect squares have odd # of factors so only perfect squares less than 1000 will end up opened.

Last perfect square which is less than 1000 is 31^2=961, so there will be toal of 31 lockers opened: 1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, ..., 31^2=961.

Answer: B (31).

shrouded1 · Answer

Note that a locker n will be touched by the student i if and only if i divides n.
All the lockers are closed initially. For a locker to remain closed it must be touched an even number of times. Or in other words, it must have an even number of factors. Conversely, if a locker number n has an odd number of factors, it will be open at the end of the operation.

Now consider any number which is not a perfect square. For any factor x, (n/x) is also a factor such that x*(n/x)=n and n/x is not equal to x. In other words, it is possible to pair up all possible factors, and hence the number always has an even number of factors.
Perfect squares on the other hand have an odd number of factors, as all factors are paired up except sqrt(n).

So the number of lockers remaining open will be the number of perfect squares less than or equal to 1000, and this is precisely given by square_root(1000) or the largest integer less than or equal to it.

Hence :   The answer is the largest integer less than or equal to  \sqrt{1000}  or 31.

nick1816 · Answer

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?
Note: The first locker that the n-th student changes is the n-th locker.

1. 500
2. 32
C. 31
D. 30
E. 250

IanStewart · Answer

If you think of, say, locker #6, it gets flipped open or closed by the 1st, 2nd, 3rd and 6th student, i.e. it will be flipped by student n any time n is a divisor of 6. So locker #6 will stay closed, because 6 has an even number of divisors. On the other hand, locker #25 say gets flipped by the 1st, 5th and 25th student, so it will end up open because 25 has an odd number of divisors.

The only numbers with an odd number of divisors are perfect squares. So the question is just asking: "How many perfect squares are there between 1 and 1000 inclusive?" Now 32^2 might be familiar, since it's a power of 2 -- it is equal to 1024, so 32 is too big, and the answer is 31.

I'd add that in a real GMAT question, when answers are simple integers, the answer choices would always be listed in increasing or decreasing order, and real GMAT questions wouldn't waste time discussing how "strange" the principal of a high school might be, so this is clearly not an official question. I suspect if something similar were to appear on the real test (which seems unlikely to me), the number of 'lockers' involved would be far less than 1000.

Bunuel · Answer

Please SEARCH before posting. Thank you.

A high school has a strange principal. On the first day, he has his st

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A high school has a strange principal. On the first day, he has his st

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards