geometry
posted by katie
At Millock Highschool, there are 1,000 lockers and 1,000 students. The first day of school, all the students line up outside of the school and enter one at a time. The first student touches evry locker,the second student touches evry locker that is a multiple of two, the third student touches evry locker that is a multiple of three, the fourth student touches evry locker that is a multiple of four, and the pattern continues. when they start all the lockers are closed. read on for it to make sence.
definition:When a student touches a locker, if it is closed they open it and if it open they close it.
1.After all the students have gone, which lockers are open and which are closed?
{15points}
2. What pattern, if any, do you notice?
{5points}
3.Explainwhy these are left open.{20points}

bobpursley
http://www.braingle.com/brainteasers/teaser.php?op=2;id=7824;comm=0

Anonymous
There is quite a bit of documentation on this problem online. Try googling "first student touches all lockers" without the quotes. I would give you some specific websites, but I'm unable to post links. Good luck!

tchrwill
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.
Respond to this Question
Similar Questions

Math
A new school has exactly 1,000 lockers and 1,000 students. the student meet outside the building and decide on this plan. The first student will enter and open al the lockers. The second student will enter and colse ever locker with … 
Math 6th
Imagine there are 1000 lockers and 1000 students at school. The 1st student who comes through the school opens every locker as they enter the school. The 2nd student walks in and closes every other locker begining with locker number … 
Math
Newon Middle School has a hall with 500 lockers, all of which are closed. Five hundred students start down the hall. The first student opens every locker. The second student closes all the lockers that are multiples of 2. The third … 
9th grade
A new high school has just been completed. There are 1,000 lockers in the long hall of the school and they have been numbered from 1 to 1,000. During lunch, the 1,000 students decide to try an experiment.  The first student, student … 
ALEGEBRA
The new school has exactly 1000 students and 1000 lockers.On the first day all the students meet outside and agree on the following plan: the first student will enter the school and open all the lockers.The second student will then … 
algebra
the new school has exactly 1000 lockers and 1000 students. on the first day of school, the students meet outside the building and agree on the following plan: the first student will enter the school and open all of the lockers. the … 
algebra
the new school has exactly 1000 lockers and 1000 students. on the first day of school, the students meet outside the building and agree on the following plan: the first student will enter the school and open all of the lockers. the … 
Math
Reasoning Word Problem: A school's lockers are numbered 1100. One hundred students enter the school one at a time. The first student opens the lockers. The second student closes the evennumbered lockers. The third student either … 
Math Word Problem(PLEASE HELP NOW)
Reasoning Word Problem: A school's lockers are numbered 1100. One hundred students enter the school one at a time. The first student opens the lockers. The second student closes the evennumbered lockers. The third student either … 
Algrebra
A new high school has just been opened, with an enrollment of 1000 students. The school has 1000 lockers, numbered 1 to 1000, to accommodate them. On the first day of school all 1000 lockers are closed. The first student enters the …