Solve High School Lockers Calculus Problem

[whisperblade]

Hi everyone, i recently got this problem from my college professor and I am either incredibly rusty or just don't know how to do this. I've tried to set up some sort of equation using f(x) but i just can't make anything fit or account for everything. any help would be appreciated.

the question follows:
A high school has 1000 students and each has a numbered locker where they keep various smelly items. Fortunately all the locker doors are shut. One by one, each student walks past the lockers, and either opens or shuts (depending on its previous position) the door of any locker that divides their own locker number. How many lockers are open at the end?

 

[hotvette]

My daughter got this same problem a few weeks back. Suggest you just plug-n-chug to map out the 1st 15-20 students until you see the pattern. At least, that's how we did it.

 

[quicknote]

Hello,

This is an interesting problem that combines both mathematics and logic. To solve this, we can use a combination of calculus and number theory.

First, let's define some variables:
- n = the number of lockers (in this case, n = 1000)
- x = the locker number of the student
- f(x) = the number of factors of x (i.e. the number of lockers that divide x)

Now, let's think about what happens when a student walks past the lockers. If the student's locker number is x, they will open or shut the door of any locker that divides x. This means that if x has an odd number of factors, the student will open the locker, and if x has an even number of factors, the student will shut the locker.

Using this information, we can create a function that represents the number of open lockers at any given time. Let's call this function g(n), where n is the number of lockers. We can express g(n) as follows:

g(n) = f(1) + f(2) + f(3) + ... + f(n)

This function represents the total number of factors of all the numbers from 1 to n. Using calculus, we can approximate this function by taking the integral of f(x) from 1 to n. This gives us the following equation:

g(n) ≈ ∫f(x)dx from 1 to n

Now, we need to find the function f(x). To do this, we can use number theory. It is a well-known fact that the number of factors of a number is equal to the number of divisors of that number. So, f(x) can be represented as the number of divisors of x.

Using this information, we can now solve the problem. We know that n = 1000, so we can plug this value into our equation for g(n) and approximate it using calculus. This will give us an estimate of the number of open lockers at the end.

However, it is important to note that this is only an approximation and may not give us the exact answer. To find the exact answer, we would need to use number theory to find the number of divisors of 1000 and then plug that value into our equation for g(n).

I hope this helps and gives you a better understanding of how to approach this problem. Good