Number Theory Proofs: Square Numbers and Irrational Roots

[randommacuser]

Hey all, I've got a few number theory exercises that are troubling me.

1. Prove a positive integer s is a square if and only if each of the exponents in its prime factorization is even.

2. Let c,d be positive, relatively prime integers. Prove that if cd is a square, c and d are squares.

3. Show that for four integers a,b,c,d, if a+b*sqrt(10)=c+d*sqrt(10), then a=c and b=d.

Hopefully someone can give me a start here. Thanks!

 

[0rthodontist]

What have you tried doing?

 

[Tide]

HINT 1: Could a number possibly be a perfect square if any of its prime factors occur an odd number of times?

HINT 2: See Hint 1.

HINT 3: Is \sqrt {10} rational?

 

[randommacuser]

I'm attempting a proof by contradiction on #1, along the lines of what Tide is hinting at. I just don't know how to show it formally.

And I know sqrt(10) is not rational, I'm just not sure how to use that yet.

 

[0rthodontist]

Can you show that if the exponents on the prime factorization of a positive number are even, then the number is a perfect square?

 

[randommacuser]

Sure, that should be the easier case. How about the other way around?

 

[0rthodontist]

Well why don't you actually do that, so that we can see.

If you do understand that then the other way around is not much harder. If a number is a perfect square then it can be written as k * k, and what can you do with each of the k's?

 

[randommacuser]

Yeah, I just had another look at #1 and it's not that difficult. Silly me...

And once I saw the reasoning #2 followed fairly easily, though I haven't quite figured out the notation.

So if anyone has suggestions for #3, I'd appreciate it!

 

[randommacuser]

Well, it's crude, but I think I have #3 as well. Thanks for all the hints, guys. If anyone is interested in how I proved any of these questions, just ask and I will try to explain as best I can.

 

[Tide]

I'm attempting a proof by contradiction on #1, along the lines of what Tide is hinting at. I just don't know how to show it formally.

And I know sqrt(10) is not rational, I'm just not sure how to use that yet.

It's really quite straightforward. Rearrange your equation to have all the rational terms on one side and all the irraitional ones on the other. Under what conditions is it possible for those two quantities be equal?