Suppose I have 2011 numbered lights in a row, all of which are initially off. Th
ID: 1941732 • Letter: S
Question
Suppose I have 2011 numbered lights in a row, all of which are initially off. Then I toggle all of the light switches, so they are all on. So then I toggle all the even number light switches so the even numbered light are on and the odd number lights are still on. Then I toggle all the lights whose number is a multiple of 3. Then I toggle all the lights whose number is a multiple of 4. The I toggle all the lights whose multiples of 5, 6, 7 etc. I keep going until the last step, when I toggle the 2011th light by itself. At the end of this procedure, which lights are on? How many lights are on and how many are off?I need help ASAP!!! Thank you.
Explanation / Answer
Suppose I have 2011 numbered lights in a row, all of which are initially off. Then I toggle all of the light switches, so they are all on. So then I toggle all the even number light switches so the even numbered light are on and the odd number lights are still on. Then I toggle all the lights whose number is a multiple of 3. Then I toggle all the lights whose number is a multiple of 4. The I toggle all the lights whose multiples of 5, 6, 7 etc. I keep going until the last step, when I toggle the 2011th light by itself. At the end of this procedure, which lights are on? How many lights are on and how many are off?
read question carefully.
first all on
second even number on and odd nubmer off
third multiple of 3 on and remainign are off
.........
light is toggled as many times as the number of divisors it has. For example, light number 24 is toggled on days 1, 2, 3, 4, 6, 8, 12 and 24. Now, divisors come in pairs like 24 = 1*24 = 2*12 = 3*8 = 4*6. So the total number of divisors is even, except when the number is a perfect square, in which case the total number of divisors is odd. Since all lights are initially off , only those lights that have an odd number of divisors have + switched on . These are light numbers 1, 4, 9, 16, 25, 36, and so on.
so 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324, and so on......
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.