Question 118643: PART C
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Imagine you are at a school that has lockers. There are 1,000 lockers, all shut and unlocked,
and 1,000 students -
HERE’S THE PROBLEM:
1) Suppose the first student goes along the row and opens every locker.
2) The second student then goes and shuts every other locker beginining with the number 2
3) The third student changes the state of every third locker beginning with number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.)
4) The fourth student changes the state of every fourth locker beginning with number 4. Imagine that this continues until the thousand students have followed the pattern with the thousand lockers. At the end, which lockers will be open and which will be closed? Why?
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Answer by scott8148(6628)   (Show Source):
You can put this solution on YOUR website! looking at a few lockers will reveal the pattern...
___ c is closed, o is open
L1 ___ o-1
L2 ___ o-1, c-2
L3 ___ o-1, c-3
L4 ___ o-1, c-2, o-4
L5 ___ o-1, c-5
L6 ___ o-1, c-2, o-3, c-6
L7 ___ o-1, c-7
L8 ___ o-1, c-2, o-4, c-8
L9 ___ o-1, c-3, o-9
L10 ___ o-1, c-2, o-5, c-10
look at the OPEN lockers ___ 1, 4, 9 ___ see a pattern? ___ WHY?
factors for numbers occur in pairs, like for 12 __ 1,12 __ 2,6 __ 3,4
___ the first factor opens and the second closes, so the locker ends up closed
but for perfect squares, one of the pairs is the same number
___ for 36 __ 1,36 __ 2,18 __ 3,12 __ 4,9 __ 6,6
___ so there is no close to go with the open
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