Locker Problem

This problem was about 1,000 lockers and each kid open a certain amount of locker or closed it or changed the pattern. Student 1 opened all of the lockers. Student 2 closes the doors of Lockers 2,4,6,8 and so on. Student 3 changes the state of lockers 3,6,9,12 and so on. Student 4 changes the state of 4,8,12,16 and so on. Student 5 changes the state of every fifth locker. Student 6 change the state of every sixth locker. With this information I must find the answer of how many lockers will be open at the end of this. This problem involves deep thinking with square numbers, prime numbers and factors. This problem is about using a lot of out learning from the past unit.

I made a picture on a whiteboard of thirty lockers and did the pattern with all of these lockers. I ended up with three lockers open, lockers 1, 4, 9, 16, 25. The connection between all of these numbers is that they are all square numbers. They were not all even numbers or odd numbers and they were not all prime numbers or composite numbers so they had to be square numbers. That basically leads you to the answer. It tells you that know you just need to find how many square numbers there are in 1,000 because the pattern must repeat.

All square numbers have an odd number of factors, which means all the other numbers have an even amount of factors. For example, 9 has the factors of 3, 9 and 1 so the locker goes open, closed and then opened. If you take a number that’s not a square number with an even amount of factors like 18, the factors are 1, 2, 3, 6, 9 and 18 so the locker goes opened, closed, opened, closed, opened and then closed again. Therefore, only square number lockers are open.

The answer is all the square numbers up to 1,000 or to be more exact, 31 lockers.

       Here is how my picture of the 30 lockers turned out :

              (The boxes represent the lockers and the crosses mean the lockers are closed.)

A observation I have from this problem is that it is just a generalization. This problem is even including our vocabulary, the word generalization. The answer was just all square numbers, which is a generalization. This is why I got very confused for a while because the answer is really just a generalization, you could make it more exact though.