Answer: So, there are 31 perfect squares between 1 and 1000. Therefore, there are 31 lockers that will remain open after all 1000 students have entered the building and changed the lockers.
Step-by-step explanation: At the beginning, all 1000 lockers are closed. Let's go step by step to see how the lockers are affected by each student:
- Student 1 enters and opens all the lockers. Now, all 1000 lockers are open.
- Student 2 enters and closes all the even-numbered lockers. This means that every second locker is closed. So now, there are 500 lockers open and 500 lockers closed.
- Student 3 changes the state of the lockers that are multiples of 3. This means that student 3 opens all the lockers that were closed and closes all the lockers that were open. So, for every third locker, its state is flipped. For example, if locker 3 was open, it is now closed. And if locker 6 was closed, it is now open.
- Student 4 changes the state of the lockers that are multiples of 4. This means that student 4 opens all the lockers that were closed and closes all the lockers that were open. For every fourth locker, its state is flipped.
- This process continues with each subsequent student, until student 1000 changes locker 1000.
Now, let's analyze the pattern:
- The lockers that have been touched by an even number of students (including zero times) will end up in their original state. For example, if locker 4 was open and is touched by students 2, 4, 8, 16, etc., it will be flipped four times and end up open.
- The lockers that have been touched by an odd number of students will end up in the opposite state from their original. For example, if locker 6 was open and is touched by students 3 and 6, it will be flipped twice and end up closed.
So, the number of students who touched a locker will determine its final state.
For any locker, only the students whose numbers are factors of the locker number will touch it. If the number of factors of a locker is even, it will end up closed. If the number of factors is odd, it will end up open.
Now, let's consider the factors of a locker. For example, let's take locker 12:
Factors of 12: 1, 2, 3, 4, 6, 12.
As we can see, the factors of 12 are 1, 2, 3, 4, 6, and 12. There are 6 factors, which is an even number. Therefore, locker 12 will end up closed.
Following this pattern, we can determine the state of each locker by counting the number of factors it has.
The only lockers that have an odd number of factors are the perfect squares (numbers that can be expressed as the square of an integer).
In the case of 1000 lockers, the perfect squares are:
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
...
31^2 = 961
32^2 = 1024 (greater than 1000)
So, there are 31 perfect squares between 1 and 1000. Therefore, there are 31 lockers that will remain open after all 1000 students have entered the building and changed the lockers.