What methods could I use to solve this problem (opening and closing lockers)?

Question

Could anyone please explain this question to me. I noticed certain patterns with the overlapping of numbers, but it all seemed too much for me to properly count. I have seen the method in the mark scheme, but I don't really understand it. In this test I have around 20 minutes to solve this question, and i'm not that good with mental maths, so brute force doesn't seem like a good option. Does anyone have a good, fast method of solving this problem? How would you approach this problem?

I would like to note that although a similar question has been asked, it does not explain the answer fully. I'm not sure what topic this would come under, so if anybody could correct the tag, it would be much appreciated. My title is also quite bad. I apologise for this.

Thank you.

Answer 1

For the first, the first student opens all the lockers. The second student closes all the even number lockers. The third student opens or closes every locker with a number divisible by $3$. You should convince yourself that every locker with a number divisible by $6$ is now open, that every locker with a number of the form $6k+3$ is now ???, that every locker with a number of the form $6n+1$ is now ???, that every locker with a number of the form $6n+4$ is now ???. The point is that $6=LCM(2,3)$ so everything repeats in sixes. For the second part, you now have to work modulo $12=LCM(2,3,4)$ Having done these, the pattern should become clear for larger numbers.