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.

Question

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.

Answer 1

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Answer 2

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Answer 3

To determine which lockers will remain open, we need to analyze the pattern of students opening and closing lockers.

Let's start with locker number 1. The first student opens it.

Next, the second student comes and closes every even-numbered locker. So, all the even-numbered lockers are now closed.

The third student reverses every third locker. Since the first student opened locker number 1, which is a multiple of 1, locker number 1 will be reversed and closed again by the third student.

The fourth student reverses every fourth locker. Locker number 1 is a multiple of 1 (first student), so it gets reversed twice, and is opened again by the fourth student.

At this point, locker number 1 has been opened, reversed, and opened again. In general, a locker will be opened and closed again when the number of students who interact with that locker is even. Lockers will remain open only if an odd number of students interact with them.

We can see that locker number 1 was opened by the first student and the fourth student. So, it has been interacted with twice, which makes the number of interactions even. Therefore, locker number 1 will be closed at the end.

Now let's consider some other lockers:

Locker number 2 - It is closed by the second student (even), reversed by the third student (odd), and not interacted with by any other student. So, locker number 2 will remain open.

Locker number 3 - It is not interacted with by the second student, reversed by the third student (odd), and not interacted with by any other student. So, locker number 3 will remain open.

We can continue this pattern of analysis for the rest of the lockers.

In general, only lockers that have an odd number of factors (including 1 and the number itself) will remain open. These lockers are called perfect square numbers because they have an odd number of factors. In this case, the perfect square numbers between 1 and 1000 are:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961.

So, the answer is that there will be 31 lockers remaining open.