The lockers @ Lucy's school are numbered from 1-1000. Lucy did a math experiment @ her school. She opened every 2nd locker. Next, starting over, she went to every third locker: if it was closed, she opend it, if it was open, she closed it. She continued this pattern; going to every 4th, 5th, 6th......1000th locker.
At the end of this experiment, are the lockers listed below open or closed? how do you know?
A) Locker #5- open or closed?
B) Locker #12-open or closed?
C) Locker #100- open or closed?
D) Locker #856- open or closed?
2006-10-19 12:47:41 · 6 answers · asked by killy 2 in Science & Mathematics ➔ Mathematics
I'll assume they all started closed.
Let's think about the lockers. A specific locker will get "toggled" for each of its factors.
A) The factors of 5 are 1 and 5. Now Lucy skipped 1, but started with touching every 2nd, then every 3rd, then every 4th, then every 5th, etc. So locker 5 only gets toggled once when Lucy is touching every 5th locker. In other words, it goes from closed to open.
#5 is open
B) The factors of 12 are 1, 2, 3, 4, 6, 12. Not counting 1, that is 5 more factors. 5 is odd, so in the end, locker #12 will be open. (Opened on 2, closed on 3, opened on 4, closed on 6 and opened on 12).
C) The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100. Not counting 1, that is 8 factors. 8 is even, so in the end, locker #100 will be closed. (Opened on 2, closed on 4, opened on 5, closed on 10, opened on 20, closed on 25, opened on 50, closed on 100).
Now let's step back for a second and think. Every number will have an even number of factors, unless it is a *perfect square* (like 49 or 100). So the answer is only the lockers that are perfect squares will be closed (1, 4, 9, 16, 25, 36, 49, etc.) With this test, you don't even have to figure out all the factors. Just see if the number is a perfect square. If yes, it is closed, if no, it is opened.
D) 856 has an even number of factors. Not counting 1, that's an odd number of toggles. 856 is *not* a perfect square, so it will be open. (The factors are 1, 2, 4, 8, 107, 214, 428, 856, in case you want to double-check.)
Summary:
A) 5 - open
B) 12 - open
C) 100 - closed (perfect square)
D) 856 - open
2006-10-19 12:57:01 · answer #1 · answered by Puzzling 7 · 0⤊ 0⤋
Locker #N is open if an odd number of the numbers 2, 3, ..., 100 evenly divide N.
A) Locker #5 is open. None of the numbers 2,3 or 4 divide 5, so it's opened when Lucy opens every 5th locker, and it's never visited again.
B) Locker #12 is open. It's opened when visiting every 2nd locker, closed when visiting every 3rd, opened when visiting every 4th, closed when visiting every 6th, and opened when visiting every 12th.
C) Locker #100 is closed
D) Locker #856 is open
2006-10-19 12:55:56 · answer #2 · answered by James L 5 · 0⤊ 0⤋
seem for the "Zoom" length someplace alongside the accurate part. i do not comprehend if this can make a distinction, yet even if it is zoomed out too a concepts, it would want to probable tutor the pages very small. not in any respect recommendations. After searching on the photo, it is no longer what's happening. It feels like the web page replaced into divided right into a grid. there is an determination to insert a spreadsheet, and probably it is what's happening. i'm at a loss what might want to reason that once you first initiate MS be conscious. Sorry. :-(
2016-12-05 00:39:34 · answer #3 · answered by Anonymous · 0⤊ 0⤋
I think you can look at what numbers are factors of the number -
5=5 - only one, so only touched once, so open
12- 2,3,4,12 - touched 4 times, so closed
100 - 2,4,5,10,20,25,50,100 - touched 8 times, an even number, so closed
856 = 2x2x2x107 - 2,2x2,2x2x2,107,2x107,4x107,856 - 7 times, so open
2006-10-19 13:06:03 · answer #4 · answered by timbo 2 · 0⤊ 1⤋
Meh...? o.O
Um....que..? @_@
2006-10-19 12:54:59 · answer #5 · answered by La Reina 3 · 0⤊ 1⤋
uhhhhh......a?
2006-10-19 12:50:26 · answer #6 · answered by BENDER IS THE BOMB!!! (Fav show) 4 · 0⤊ 1⤋