A new high school has just been completed. There are 1,000 lockers in the long hall of the school and they have been numbered from 1 to 1,000. During lunch, the 1,000 students decide to try an experiment.
- The first student, student 1, runs down the row of lockers and opens every door.
- Student 2 closes the doors of lockers 2, 4, 6, 8 and so on to the end of the line.
- Student 3 changes the state of the doors of lockers 3, 6, 9, 12 and so on to the end of the line. (the student opens the door if it is closed and closes the door if it is opened)
- Studnet 4 changes the state of the doors 4, 8, 12, 16 and so on. Student 5 changes the state of every fifth door, student 6 changes the state of every sixth, and so on until all 1000 students have had a turn.
When the students are finished, which lockers doors are open?
There won't be a pattern. Take the factors of each number, include 1 and the number itself. Count the number of factors. This is the number of "state" flips. If it is even, the locker is closed. If it is odd, the locker is open.
@bobpursley
so you mean we have to do a factor tree for 1000 numbers?!
Yeah, that's the tkciet, sir or ma'am
Side of a triangle are in the ratio 2:3:4 if shortest side is 45mm find the length of the the other sides
To determine which lockers' doors are open after all the students have had a turn, we need to analyze the pattern and behavior of each student. Let's break down the problem:
First, we need to initialize the lockers' state. At the beginning, all the lockers are closed.
Next, we need to consider each student's action and how it affects the lockers:
- Student 1: Opens all the lockers.
- Student 2: Closes every second locker (2, 4, 6, 8, ...).
- Student 3: Changes the state of every third locker. If a locker is closed, they open it, and if it is open, they close it. (3, 6, 9, 12, ...).
- Student 4: Changes the state of every fourth locker. Same as student 3, if a locker is closed, they open it, and if it is open, they close it. (4, 8, 12, 16, ...).
This process continues until all 1,000 students have had a turn, each changing the state of the lockers according to their pattern.
To find out which lockers end up open, we can track the number of times each locker is operated on or "toggled." If a locker is toggled an odd number of times, it will eventually end up open, and if it is toggled an even number of times, it will end up closed.
Now let's calculate the state of each locker at the end:
Lockers 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 (perfect squares) are the ones toggled an odd number of times.
Therefore, the doors of these lockers will be open at the end.