I have a math question that goes like this: There are 500 light that have on and off switches. One person goes and turns all the lights on. The second person switches the lights from 2,4,6,8...and so on. THe 3rd person switches 3,6,9,12....and so on. This goes on for 500 people. How many lights will be left on in the end.

Question

I do not even know how to start this problem please someone help

Try to find a pattern with even-numbered people and odd-numbered people.

I think that it has something to do with series but Im not sure

I tried to find a pattern and can't

You just look at a particular light nr. n and see how many times is is swithced on and off. Now every number has a factorization in terms of prime numbers. We can formally denote the prime numbers by:

p1, p2, p3 etc, where

p1 = 2

p2 = 3

p4 = 5 etc.

Any number n is of the form:

n = p1^k1 * p2^k2 * p3^k3 * ...

e.g. 14 = 2*7 =

2^1 * 3^0 * 5^0 * 7^1 * 11^0*... =

p1^1* p2^0 * p3^0 * p4^1 * p5^0*...

What happens to light nr. n of the form

n = p1^k1 * p2^k2 * p3^k3 * ...

The only persons that can affect light nr. n are persons that have a number r such that:

r = p1^x1 * p2^x2 * p3^x3 * ...

such that xi <= ki

for all i.

because r must be a divisor of n.

How many values for r are there? Each xi has to be in the range from zero to ki, so you have:

(k1+1)*(k2+1)*(k3+1)*...

on and off switchings.

Now you know that a product of numbers is even if one or more factors in the product are even. The only way a product can be odd is if all the factors are odd. Now, you need to switch the light on and off an odd number of times to get it switched on.

This means that all the ki+1 must be odd for the lights that are left on in the end. So, the ki must be even. The prime factorization for the numbers of the lights that are left on thus contain only even powers of prime numbers, which means that they are squares.

So, al the squares (including 1) will be left on. Since sqrt[500] = 22.36 there are 22 lights that are left on:

1^2, 2^2 = 4, 3^2=9,...,22^2= 484

maybe you should just give up? or ask your teacher? or just miss it out

If you are having problems than you should stay after school and ask your teacher for help. Or you can go before school starts. But if that doesn't work you can ask you teacher to give you extra help! Hope this works! Good Luck!!!!!!