Reasoning Word Problem: A school's lockers are numbered 1-100. One hundred students enter the school one at a time. The first student opens the lockers. The second student closes the even-numbered lockers. The third student either closes or opens every third locker. The remaining students continue the pattern. After all the students have passed the lockers, which lockers are open.

o_o ummmm what????? HELP ME PLEASE!!

51 are open 49 are Closed?

Sure, I can help explain the reasoning behind this word problem. Let's break it down step by step.

First, we know that there are 100 lockers numbered from 1 to 100. The first student opens all the lockers initially.

The second student comes along and closes every even-numbered locker. So after the second student, all the even-numbered lockers are closed.

Now, the third student follows a different pattern. They go to every third locker and either open it if it's closed or close it if it's open. So, for example, locker number 3 will be opened by the third student, while locker number 6 will be closed.

The remaining students continue this pattern. The fourth student will go to every fourth locker, the fifth student to every fifth locker, and so on.

To see which lockers remain open after all the students have passed, we need to observe the pattern. Notice that the lockers will only be touched by students whose numbers are factors of the locker's number. So, for example, locker number 12 will be touched by students 1, 2, 3, 4, 6, and 12.

Now, let's consider the factors of each locker number. If a locker has an odd number of factors, it will end up being open at the end. Why is this? Well, every time a student touches a locker, it changes its state (from open to closed or vice versa). So, for even factors, the locker will be touched and its state will switch twice, resulting in it reverting back to its original state of being closed. However, for odd factors, the locker's state will switch an odd number of times, leaving it open at the end.

Using this reasoning, we can find that only the lockers with perfect square numbers (such as 1, 4, 9, 16, etc.) will end up being open after all the students have passed.

So, to answer the question, the open lockers will be locker numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.