Open Lockers

The only lockers that remain open are perfect squares (1, 4, 9, 16, etc) because they are the only numbers divisible by an odd number of whole numbers; every factor other than the number's square root is paired up with another. Thus, these lockers will be 'changed' an odd number of times, which means they will be left open. All the other numbers are divisible by an even number of factors and will consequently end up closed. So the number of open lockers is the number of perfect squares less than or equal to one thousand. These numbers are one squared, two squared, three squared, four squared, and so on, up to thirty one squared. (Thirty two squared is greater than one thousand, and therefore out of range.) So the answer is thirty one.

1 is visited by the first student

2 is visited by the first and second student

3 is visited by the first and third student

4 is visited by first, second and fourth student.

notice that they are visited by their factors. all factors are paired by another unless the factor is repeated which occurs when the number is a square number.

hence all numbers are visited an even number of times other than square numbers. When a locker is visited an even number of times it is closed, when it is visited an odd number of times it is opened. Largest square number smaller than 1000 is 31*31=961. So 31 lockers are opened

Note that any locker that is open has been opened or closed an odd number of times. This means that all open lockers must have an odd number of factors.

The number of factors any given number has can be attained the following way:

Rewrite the number as (a^b)x(c^d)x(e^f)x(g^h)x…. where a,c,e,g, etc. are the numbers prime factors.Thus, the number of as the number can have in any given factor is some number between 0 and b, inclusive, and the number of c's a number can have are between 0 and d inclusive, etc. So, therefor the number of factors any number has is (b+1)x(d+1)x(f+1)x… Clearly, since this number must be odd, all of b,d,f, etc. must therefore be even, and thus the number must be a square.

Since we know that a locker is open if and only if it is a square, and that there are 31 squares less than or equal to 1000, there must be 31 lockers open.

If you examine the first 20 or so lockers and determine whether they will be open or closed, you will find a pattern. This is 1 open followed by 2 closed ,then 1 opened followed by 4 closed and 1 opened followed by 6 closed and so on. So, locker 1 is open, 2-3 are closed, 4 is open, 5-8 are closed, 9 is open, 10-15 are closed and so on. Square numbers are all open. There are 31 square numbers upto 1000 and hence 31 lockers are open.

• Lockers with a number with an odd number of divisors will be open in the end
• Lockers with a number with an even number of divisors will be shut in the end
• The number of divisors of a number N with prime representation $N = p_1^{m_1}× p_2^{m_2}×... ×p_n^{m_n}$ is equal to $(m_1+1)(m_2+1)...(m_n+1)$ .

Corollary: If the prime representation of a number contains an odd power, then that number has an even number of divisors; if the prime representation of a number contains only even powers, then that number has an odd number of divisors.

Furthermore: The prime representation of N has all powers even $\Leftrightarrow$ N is a square

Conclusion: Exactly the lockers with a square as their number will be open in the end. The largest square upto 1000 is $31^2$ , so there will be $\boxed{31}$ lockers open

All lockers are closed at the start Student opens all lockers The 2nd opens every second locker.

Ect.

Lockers are opened/closed if the nth student is going and they have a factor of n. The trick is that for each factor, there is another factor and the pair multiply to get the original number. This means the lockers will return to closed, except...

Squares have an odd number of factors, as their square root squared is...

Well, the original number, meaning the pair of factors is really a pair of two of the same number.

This means the squares will be open at the end. There are 31 squares below 100, so that is the answer

Being a slob, I just wrote it all out for the first 70 lockers. Noticed that, indeed, lockers 1, 4, 9, 16, etc. we’re open. Specifically, I noticed that the number of sequential closed lockers were 3, 5, 7, 9, etc. which pointed out the series. Not elegant, but effective.

simply count the length of the list of divisors. if it is odd, the locker will be open. only perfect squares qualify.

For any natural number $n$ ,

$n$ is a perfect square $\iff$ $n$ has an odd number of positive divisors (including $1$ and itself).

a very good question

The lockers that are open have odd no of divisors => perfect squares. Therefore #perfect squares = $\left \lfloor \sqrt{1000} \right \rfloor - 1 + 1 = \left \lfloor \sqrt{1000} \right \rfloor = 31$

That is, the kth student opened/closed the nth locker exactly when k is a factor of n. This holds for all n, so that the kth student opens/closes the nth locker exactly when k is a factor of n.

We can pair up factors of n, so that k pairs up with n/k, whenever k and n/k are different. That is, if the kth student opens the locker, then the n/k'th student closes it, and vice versa. The only time, then, that the door may be open at the end is when n/k = k for some k. This means n = k^2 is a perfect square.

So how many perfect squares are there between 1 and 1000? that is 31.

see it is a very typical question but just we have to look into the divisors of a number. a number which has an odd number of divisors will remain opened because it is open , close , open ,............... so every number having an odd divisor will be opened. a prime no has 2 divisors so it can't be a prime numbered locker. `so first how to find the total no of divisors of a number here you can learn : https://brilliant.org/discussions/thread/divisors-of-an-integer/

in every a case of a single prime raised to an even power the no of divisors is always odd and in case where there are n primes in multiplication raised to the power which is even are also perfect squares. and yield odd no of divisors. so only perfect squares have odd no of divisors so they are the only lockers which will be open and so there are 31 perfect squares so the answer is 31.