Suppose you're in a hallway lined with 100 closed lockers.
You begin by opening every locker. Then you close every second locker. Then you go to every third locker and open it (if it's closed) or close it (if it's open). Let's call this action toggling a locker. Continue toggling every nth locker on pass number n. After 100 passes, where you toggle only locker #100, how many lockers are open?
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Answer:
10 lockers are left open
Step-by-step explanation:
The number of toggles of a door determines the final status. If a door is toggled for odd times, it will end up with open, whereas if even time, then the door will be closed at the end.
Thus,
For door 1, it will be toggled once therefore it will be open.
For door 2, it will be toggled twice (in pass 1 and 2) and will be closed.
For door 3, it is toggled twice (pass 1, 3) and will be closed.
For door 4, it is toggled 3 times (pass 1, 2, 4) and will be open.
For door 100, it is toggled 9 times (pass 1, 2, 4, 5, 10, 20, 25, 50 and 100) and ends up open.
Therefore, it can be reduced to find the numbers that have odd number of factors between 1 and itself. Locker numbers 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 have this property. This is so as all the other numbers have even number of factors that can multiply to itself while the mentioned numbers have one more that is the square root of itself. Therefore, the perfect squares are left open.
Step-by-step explanation:
Answer: 10 lockers are left open:
Lockers #1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Each of these numbers are perfect squares. This problem is based on the factors of the locker number.
Each locker is toggled by each factor; for example, locker #40 is toggled on pass number 1, 2, 4, 5, 8, 10, 20, and 40. That's eight toggles: open-closed-open-closed-open-closed-open-closed.
The only way a locker could be left open is if it is toggled an odd number of times. The only numbers with an odd number of factors are the perfect
squares. Thus, the perfect squares are left open.
For example, locker #25 is toggled on pass number 1, 5, and 25 (three toggles): open-closed-open.