Problem Of The Week Help!?

Question

A school corridor is lined with 1000 lockers, all closed. There are
1000 students who are sent down the hall in turn according to the following rules.
The first student opens every locker. The second student closes every second
locker, beginning with the second. The third student changes the state of every
third locker, beginning with the third, closing it if it is open and opening it if
it is shut. ( this happens for all 1000 students )The problem is: Which
lockers remain closed after all the students have marched?




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Can you tell me the answer and how you got it... I know for sure the answer is 31 cause my teacher told me but how is it 31??? please help me...

Answer 1

If a number is a perfect square it will have an odd number of factors,
e.g 4 has factors 1, 2, 4, whereas all other numbers have an even
number of factors. If a particular locker is visited an odd number of
times it will be open at the end of the procedure you describe;
otherwise it will be closed. So the open lockers are numbered:

1, 4, 9, 16, 25, 36, ..... 900, 961

The last number is 31^2 so 31 is the number of lockers left open.

To show that square numbers always have an odd number of factors,
consider a square like 36. This can be put into prime factors as
2^2 x 3^2. Note that all its prime factors will be raised to EVEN
powers since it is a perfect square.

Now the factor 2 can be chosen in 3 ways, i.e. not at all, once,
twice. And factor 3 can also be chosen in 3 ways, i.e. not at all,
once, or twice.

(If neither is chosen we get the factor 2^0 x 3^0 = 1).

Altogether there will be 3 x 3 = 9 factors of 36. These are:

1, 2, 3, 4, 6, 9, 12, 18, 36

Now the important point was made earlier that the prime factors of a
perfect square are always raised to some even power, so we could have

a^2 x b^4 x c^2 where a, b, c are primes.

In this example a could be chosen in 3 ways 0 times, once or twice.
b could be chosen in 5 ways 0 times, 1, 2, 3, 4 times
c could be chosen in 3 ways 0 times, 1, 2, times

So altogether the complete number will have 3 x 5 x 3 = 45 factors.

For any number that is not a perfect square there will ALWAYS be an
even number of factors

e.g. a^3 x b^2 x c will have 4 x 3 x 2 factors = 24 factors

If it is not a perfect square at least one of its prime factors will
be raised to an ODD power, and that means the factor can be chosen in
an EVEN number of ways, ensuring that overall there will be an even
number of factors.

The general rule for the number of factors is to increase the powers
of the factors by 1 and multiply these together.

So a^n x b^m x c^p will have (n+1)(m+1)(p+1) factors.

2^3 x 5^4 x 7^2 will have 4 x 5 x 3 = 60 factors

Answer 2

First of all I don't know how she got 31 unless she rounded it off.
1000 divided x 333.3 =3000.3000300030003
then 333.3 x 3=999.9 to verify .I hope this is it!
Maybe someone else can give a better answer.
Good Luck!
I just read Dannyboys answer He is So right.
Open his link .