There is a long row of doors labeled 1 through 10000 Originally, they are all cl

Question

There is a long row of doors labeled 1 through 10000 Originally, they are all closed. The first person opens all of them. The second person closes the 2nd, the 4th, the 6th, etc. The third person changes the status (that is, opens a closed door and closes an open door) of the 3rd, the 6th, the 9th, etc. This continues, with the nth person changing the status of the doors that are multiples of n until the 10000th person changes the status of the 10000th door. When this is finished, which doors are open and which are closed? show all work

Explanation / Answer

let the first 10 students go do their open/ shut thing with the lockers. The students who come after them are not going to touch lockers 1-10, so we can see which ones in that first batch are still open and try to guess the pattern When we do that, we find that lockers 1, 4, and 9 are open and the others are closed. Now, that isn't much to go on, so maybe you could let the next 10 students go do their thing. Then the first 20 lockers are through being touched, and we find that lockers 1, 4, 9, and 16 are the only ones in the first 20 that are still open. So what is the pattern? Let's take any old locker, like 48 for example. It gets its state altered once for every student whose number in line is an exact divisor of 48. Here is a chart of what I mean: this Student leaves locker 48 1 open 2 shut 3 open 4 shut 6 open 8 shut 12 open 16 shut 24 open 48 shut Notice that 48 has an even number (ten) of divisors, namely 1,2,3,4,6,8,12,16,24,48. So the locker goes open-shut-open-shut ... and ends up shut. Any locker number that has an even number of divisors will end up shut. Which numbers have an odd number of divisors? That's the answer to this problem. Just to help you along, here are the locker numbers up to 100 that are left open: 1,4,9,16,25,36,49,64,81,100. See if you can describe these numbers in a different way from "having an odd number of divisors." Think about multiplying numbers together. When you understand how to describe them, you will see that 31 of the 1000 lockers are still open

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