Problem

[Countolaf]

Choose the correct answer:

"A school has 1000 doors and 1000 students. Initially all doors are closed. A student goes around the school and opens every door. Another student closes every second door. The next student changes the state of each third door - if it is open he closes it, if it is closed he opens it. And so on. The N-th student changes the state of all N-th doors."

(I) 31 doors will stay open at the end.
(II) Students whose number is prime will close just one door.
(III) Students whose number is a perfect square will open just one door.
(IV) The last door to be open will be 961.

a) All affirmations are false.
b) Just one affirmation is false.
c) Two affirmations are false.
d) All affirmations are correct.
e) Just one affirmation is correct.

[Fox Mc Cloud]

a) all are false

(I) 31 doors will stay open at the end. --> I doubt that
(II) Students whose number is prime will close just one door. --> no they close every door a prime nrs away froù the prime door too.
(III) Students whose number is a perfect square will open just one door. --> lol?
(IV) The last door to be open will be 961. --> all doors are closed and door 1000 will be opened as last.

[Countolaf]

The correct answear is c

[Fox Mc Cloud]

Can you explain why?

[Countolaf]

First consideration:
Any natural number can be represented as (p^a)(q^b)......(r^c), being p, q, r, ......, prime numbers and a, b, c, ......, natural numbers.
Second consideration:
A natural number in the form (p^a)(q^b)......(r^c) jas a total of+ (a+1)(b+1)......(c+1) different natural numbers it can be divided by them. The only possibility that a natural number can be divided to a total of an uneven number of numbers is if a, b, c, ......, are all even numbers, so (a+1)(b+1)......(c+1) is an uneven number. Then the number has to be a perfect square.
Third consideration:
The doors which will be kept open are the ones which can be divided to a total of an uneven number of numbers. So the doors which number is a perfect square will be kept open.
Fourth consideration:
The natural numbers between 1 and 1000 which are perfect squares of natural numbers are 1^2, 2^2, 3^2, ......, 31^2.
So:
(I) TRUE, there will be 31 doors open.
(II) FALSE, they can close more than one door. Take as an example 4. The student number 4 will close doors 12, 24 and others.
(III) FALSE, they will open more than one door. Take, again, the example of the student number 4. He will open door 8, 16 and others.
(IV) 31^2 = 961, so the last door which will be left open is number 961.

Counting, we have 2 false and 2 true affirmations, so "c) Two affirmations are false." is the correct option.