Harder question:
Binary perfection. Can a number consisting of 300 ones and some number of zeros be a perfect square? Explain your answer.

Question

turingtest · Answer

moneybird taught me the trick on these...

anonymous · Answer

Explain pls.

turingtest · Answer

you mean the answer?
No, the problem hasn't been up long enough.

anonymous · Answer

Pumas campeón!

anonymous · Answer

Thinking....

turingtest · Answer

I'm gonna see if Jyqft has something doable this time...

anonymous · Answer

No-idea

turingtest · Answer

It was destined for meta-math anyway, but I know moneybird will solve it if I put it there...

anonymous · Answer

This number is divisible by 3 (because of the divisibility rule) but not by 9. Therefore it cannot be a perfect square. :)

turingtest · Answer

Yeah TracysaurusRex, that's it!
I didn't know about that rule until I came on this site.

anonymous · Answer

wow that is very nice rule,

turingtest · Answer

moneybird referenced the rule used above as a special case of this theorem, which I had never heard of either: http://answers.yahoo.com/question/index?qid=20080716092006AAWNZNu

turingtest · Answer

Not sure I see the connection though, but I'm sure it's insightful.

anonymous · Answer

i think that rule is simply because of the fact that a perfect square must have only 2 factors 1 and the number itself,

and if a number is a multiple  of threee,
3 must come in its prime factorization,

and it must come even number of times,
hence it hasto be a multiple of nine to be a perfect square,

as far as that theorem is concerned, 
it is elegant

mathmate · Answer

Stom, you were probably thinking of primes!

A perfect square has to have an ODD number of factors, like 100 has 1,2,5,10,20,50,100.

anonymous · Answer

what i meant was a perfect squares must have only 2 factore 1 and the number itself,

the number itself can have its own factors as well,

thats not a n issue at all,

mathmate · Answer

Just in case it's relevant, is the number in base 10 or base 2?

turingtest · Answer

It sounds like it should be binary now that you mention it, but the answer is what TracySaurus gave, so it must be base 10 I suppose.

turingtest · Answer

By "binary" they must just mean the fact they are using two numbers, not as a base.

mathmate · Answer

Oh great, so that was the official answer, and it makes sense too.