Question 61844: My daughter has this as an extra credit question and she could really use the points and I am lost in helping her. Will someone please help me and explain this?
A very long hallway has 100 lights with pull cords hanging from the ceiling. All the lights are off. A person comes along and pulls every cord, turning on all the lights. A second person comes along and pulls every second cord. This would turn off light numbers 2, 4, 6 etc. A third person comes along and pulls every third cord. Turning on some lights and turning off others. This continues until the one hundredth person comes along and pulls the one hundredth cord. At this point, which lights are on?
hint- 10 lights total are on.
Now they are looking for which cord number are on.
Found 2 solutions by joyofmath, brandonpark2889:
Answer by joyofmath(189)   (Show Source):
You can put this solution on YOUR website! The lights that are on are the ones corresponding to perfect squares:
1 4 9 16 25 36 49 64 81 and 100.
Here's why:
Pick a number, say 20. Look at the divisors of 20, the numbers that divide evenly into 20. They are 1, 2, 4, 5, 10, and 20. There are 6 divisors. If you turn the light on, then off, then on, then off, then on, then off you've pulled the cord 6 times, corresponding to the 6 divisors. Any number having an even number of divisors will have the light eventually turned off because you've turned the light on then off some number of times.
The reason divisors have to do with pulling the cord is that, for example with bulb number 20, multiples of 1 will touch that bulb (i.e. 1,2,3...,19,20). Multiples of 2 will touch that bulb (2,4,6,8,10,12,...20). Multiples of 4 will touch that bulb (4,8,12,16,20). Multiples of 5 will touch that bulb (5,10,15,20). Multiples of 10 will touch that bulb (10,20). And multiples of 20 will touch that bulb (20). So, bulb #20 is pulled 6 times, corresponding to its 6 divisors.
However, if you have an odd number of divisors then the light will stay on. For example, take the number 16. Its divisors are 1, 2, 4, 8, and 16. It has 5 divisors. So, the light goes on, off, on, off, on. The light stays on.
So, numbers with an even number of divisors leave the light off and numbers with an odd number of divisors leave the light on.
So, the question is, why do perfect squares have an odd number of divisors while other numbers have an even number of divisors?
Part of the answer has to do with the fact that divisors come in pairs in non-squares. The number 20 has the divisors 1, 2, 4, 5, 10, and 20. Notice that 1 and 20 pair up and that 1x20 = 20. Notice that 2 and 10 pair up and that 2x10 = 20. Notice that 4 and 5 pair up and that 4x5 = 20. So, if x is a divisor of 20 so is 20/x. E.g. 4 is a divisor of 20 and so is 20/4. 10 is a divisor of 20 and so is 20/10.
Now, let's look at perfect squares. The number 16 has divisors 1, 2, 4, 8, and 16. The numbers pair up: 1 and 16, 2 and 8. But, there's an extra number. It doesn't pair up with anything. It's the square root of 16, which is 4. So, 1x16=16, 2x8=16, and also 4x4=16. But, we don't count a divisor twice. So, in perfect squares there are an odd number of divisors. So, the lights for perfect square numbers stay on!
Make sense?
Answer by brandonpark2889(31) (Show Source):
You can put this solution on YOUR website! The lights that are on are the ones corresponding to perfect squares:
1 4 9 16 25 36 49 64 81 and 100.
Here's why:
Pick a number, say 20. Look at the divisors of 20, the numbers that divide evenly into 20. They are 1, 2, 4, 5, 10, and 20. There are 6 divisors. If you turn the light on, then off, then on, then off, then on, then off you've pulled the cord 6 times, corresponding to the 6 divisors. Any number having an even number of divisors will have the light eventually turned off because you've turned the light on then off some number of times.
The reason divisors have to do with pulling the cord is that, for example with bulb number 20, multiples of 1 will touch that bulb (i.e. 1,2,3...,19,20). Multiples of 2 will touch that bulb (2,4,6,8,10,12,...20). Multiples of 4 will touch that bulb (4,8,12,16,20). Multiples of 5 will touch that bulb (5,10,15,20). Multiples of 10 will touch that bulb (10,20). And multiples of 20 will touch that bulb (20). So, bulb #20 is pulled 6 times, corresponding to its 6 divisors.
However, if you have an odd number of divisors then the light will stay on. For example, take the number 16. Its divisors are 1, 2, 4, 8, and 16. It has 5 divisors. So, the light goes on, off, on, off, on. The light stays on.
So, numbers with an even number of divisors leave the light off and numbers with an odd number of divisors leave the light on.
So, the question is, why do perfect squares have an odd number of divisors while other numbers have an even number of divisors?
Part of the answer has to do with the fact that divisors come in pairs in non-squares. The number 20 has the divisors 1, 2, 4, 5, 10, and 20. Notice that 1 and 20 pair up and that 1x20 = 20. Notice that 2 and 10 pair up and that 2x10 = 20. Notice that 4 and 5 pair up and that 4x5 = 20. So, if x is a divisor of 20 so is 20/x. E.g. 4 is a divisor of 20 and so is 20/4. 10 is a divisor of 20 and so is 20/10.
Now, let's look at perfect squares. The number 16 has divisors 1, 2, 4, 8, and 16. The numbers pair up: 1 and 16, 2 and 8. But, there's an extra number. It doesn't pair up with anything. It's the square root of 16, which is 4. So, 1x16=16, 2x8=16, and also 4x4=16. But, we don't count a divisor twice. So, in perfect squares there are an odd number of divisors. So, the lights for perfect square numbers stay on!
Make sense?
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This person makes PERFECT SENSE. Watch. The reason squares have the odd divisors is that the squares have the same number multiplied, such as 4x4=16. Which means that it wouldn't have even. Think of all the divisors of 16. 1,2,4,8,16. See? 4 is in the middle, and that is because the 4 is squared. Now compare this with non-squared number, 50. Think of all of them. 1,2,5,10,25,50. See? it has even divisors because there is no squared numbers. 5 x 10 is the middle, but they are not squared. Do you get it?
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I like math, period.
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