There are 100 lockers and 100 kids....calling all math whizzes! Can you help me?

Question

There are 100 lockers and 100 kids. All the lockers are closed.
The first kid opens all the lockers.
The second kid closes lockers #2, 4, 6, ..... etc.
The third kid only looks at lockers #3,6,9,12 ...etc. If the locker is open she closes it, if it's closed she opens it.
The fourth kid only looks at lockers #4, 8, 12, 16...etc. Again, like kid #3, if the locker is open she closes it, if it was closed he opens it.
This procedure continues until 100 kids have passed the 100 lockers.

a) How many lockers and which ones (like 7, and 33...etc) were open at the end?

b0figure out the pattern.

c) If we now have 1000 lockers and 1000 kids, how many lockers and which ones would still be open?

d) Of those 1000 lockers that are open at the end, WHY are they open?



please help me w/ this problem!
It's been stumping me foreverrr!
if you're good at math please answer the four questions a through d and explain it to me cause i don't GET IT!

grr...help!

Answer 1

The square numbers will be open,
1,4,9,16,25,36,49,64,81,100, so 10 of them.
This is because only square numbers have an odd number of factors including 1. Only an odd number of openings and closings will result in a door being open.
For 1 to 1000, the squares of 1 through 31 are less than 1000, so 31 of them will be open.

Answer 2

If the number of the locker has an even number of factors, it will get closed as many times as it was open - so in the end, it will be closed. If the number of factors is odd - it will stay open.

To give an example:

locker #1 - opened by the first kid and left alone after that - 1 has 1 factor - stays open.

locker #2 - opened by first kid and closed by second - 2 factors - closed in the end.

locker #6 - opened by 1, closed by 2, opened by 3, closed by 6 - closed (4 factors).

locker #9 - opened by 1, closed by 3, opened by 9 - open (3 factors).

Most numbers have an even number of factors. Prime numbers, for example, have exactly 2 factors. The only kind of numbers with odd number of factors are squares:

1 - 1
4 - 1, 2, 4
9 - 1, 3, 9
16 - 1, 2, 4, 8, 16
25 - 1, 5, 25

etc.

so, the lockers that will remain open are the ones whose number is a square. Up to 100, there's 10 like that. up to 1000 - 31.

Answer 3

A locker will be open or closed by a kid if that kid's number is a factor of the locker's number.

For example, locker #12 will be handled by kids # 1, 2, 3, 4, 6, 12.

So a locker will remain open if its number has an odd number of factors.

Now think: what numbers have an odd number of factors? Certrainly not primes! (All primes have exactly two factors, 1 and themselves.)

Answer 4

a) The following lockers will be closed:
2 3 5 7 11 13 17 19 23
29 31 37 41 43 47 53 59 61 67
71 73 79 83 89 97

b) Prime numbers will be closed - A prime is only divisible by itself and 1. Therefore, all lockers are opened by kid #1. Primes will only be visited once more, when that associate kid visits it. That kid will then close it and it will not be visited again.

c) Same thing. Just get the prime numbers up to 1000

d) same reason as B

Answer 5

The numbers with odd factors will be open in the end.

So, technically, whatever be the number of kids, locker no. 1 will be open.

Even for prime numbers, kid 1 opens the prime number (say 7) and kid 7 has to close locker 7.

Answer 6

The prime numbers will be open at the end.