there are lockerslined up numbered 1 to 1000.They are all closed . i walk by and open all the lockers . then Al walks by and goes to every second locker starting at #2and closes it. Then Mary goes to every third locker starting with #3 closing the opeaned lockers and opening the closed lockers. The n Jose walks by and goes to every fourth locker starting at #4 closing the opened locker and openining the closed lockers. this rutine goes on untile eviline goes to locker 1000 and closes it . After this is over which lock ers will be open?why?

I don't remember the answer and I'm not necessarily going to sit down and figure this out. BUT....I will give you a hint I THINK is a correct hint:

Think about prime numbers.

I apologize if it's not. But my dad gave me this question (or a similar one) years ago and I think I remember that being part of the answer.

Matt

To determine which lockers will be open at the end, we need to analyze the pattern and understand the effect of each person's actions.

Let's break it down step by step:

1. When I walk by, I open all the lockers. So all the lockers are initially open.

2. When Al walks by, he starts at locker #2 and closes every second locker. This means all even-numbered lockers (2, 4, 6, etc.) will be closed.

3. When Mary walks by, she starts at locker #3 and follows the same pattern as Al but with every third locker. So, she will close locker #3, which was initially open, and open it if it's closed.

4. When Jose walks by, he starts at locker #4 and follows the same pattern as Al but with every fourth locker. So, he will close locker #4, which was initially open, and open it if it's closed.

The pattern continues like this with each subsequent person closing or opening lockers based on a specific interval.

Now, let's consider the effect of each person's actions on the lockers:

Al, Mary, Jose, and Eviline follow the same pattern as they walk by. They target lockers based on their position in the sequence and close or open them accordingly.

To simplify our analysis, let's look at the effect of each person's actions on a locker by focusing on its factors.

- Locker #1: This locker will be opened once by me and not closed by any subsequent person. So it remains open at the end.

- Locker #2: This locker will be opened once by me, closed by Al, and not affected by any subsequent person. So it remains closed at the end.

- Locker #3: This locker will be opened once by me, closed by Al, reopened by Mary, and not affected by any subsequent person. So it remains open at the end.

By analyzing the patterns like this for each locker, we observe that lockers with an odd number of factors (including 1 and the number itself) will end up open, while lockers with an even number of factors will end up closed.

Prime numbers have only two factors (1 and themselves), so they will remain open as they are not closed by any subsequent person.

Hence, the lockers that will be open at the end are the ones with numbers corresponding to prime numbers between 1 and 1000.

To determine which specific lockers, you can employ a prime number generator or search for a list of prime numbers up to 1000.

I hope this explanation helps you understand the process and find the solution to the problem!