## Number theory questions

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!
 PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
 Recognitions: Science Advisor What have you tried doing?
 Recognitions: Homework Help Science Advisor 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?

## Number theory questions

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.
 Recognitions: Science Advisor Can you show that if the exponents on the prime factorization of a positive number are even, then the number is a perfect square?
 Sure, that should be the easier case. How about the other way around?
 Recognitions: Science Advisor 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?
 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!
 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.

Recognitions:
Homework Help