the new school has exactly 1000 lockers and 1000 students. on the first day of school, the students meet outside the building and agree on the following plan: the first student will enter the school and open all of the lockers. the second student will then enter the school and close very locker with an even number. the third student will 'reverse' every third locker. the fourth student will reverse every fourth locker; and so on until all 1000 students have entered the building and reversed the proper lockers. which lockers will finally remain open? SHOW ALL WORK. explain your answer.

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To find out which lockers will finally remain open, let's go through the process step by step:

Step 1: The first student opens all 1000 lockers.

Step 2: The second student closes every even-numbered locker. So, they will close lockers 2, 4, 6, 8, and so on until locker 1000.

Step 3: The third student 'reverses' every third locker. This means that if a locker is closed, the student will open it, and if a locker is open, the student will close it. The student will reverse the status of lockers 3, 6, 9, 12, and so on until locker 999.

Step 4: The fourth student 'reverses' every fourth locker. This means that the student will change the status of the lockers, similar to what the third student did. The student will reverse the status of lockers 4, 8, 12, 16, and so on until locker 1000.

This process continues until all 1000 students have entered the building and reversed the proper lockers.

Now, let's analyze which lockers will finally remain open:

For a locker to remain open, it must have been changed an odd number of times: once when it was opened initially and then an odd number of times during the later student processes.

A locker will only be changed during a student's turn if the student's number is a factor of the locker number. For example, student 5 will change locker number 15 because 5 is a factor of 15.

To determine if a locker's final status will be open, we need to determine the number of factors it has.

For example, consider locker 16. The factors of 16 are 1, 2, 4, 8, and 16. It means that locker 16 will be changed during the first student's turn, the second student's turn, the fourth student's turn, and finally the sixteenth student's turn. So, locker 16 will be changed an even number of times and will end up closed.

To remain open, a locker must have an odd number of factors. This is only possible if it is a perfect square. Perfect squares have an odd number of factors because they have extra factors in common, known as square factors.

For example, consider locker 25. The factors of 25 are 1, 5, and 25. Locker 25 will be changed during the first student's turn, the fifth student's turn, and the twenty-fifth student's turn. Since locker 25 has an odd number of factors, it will remain open.

Following this pattern, the lockers that will remain open are the lockers that are perfect squares between 1 and 1000 (inclusive).

To find out how many perfect squares are there between 1 and 1000, you can take the square root of 1000 (approximately 31.62), which is the largest perfect square less than or equal to 1000. So, there are 31 perfect squares between 1 and 1000.

Therefore, 31 lockers will remain open in total.

You can also list down the specific locker numbers that will remain open by finding the square of numbers 1 to 31:

1^2 = 1
2^2 = 4
3^2 = 9
...
31^2 = 961

So, the lockers with numbers 1, 4, 9, 16, 25, ..., 961 will remain open.