Proving Perfect Squares: A Study in Number Theory

[dancergirlie]

Homework Statement

If C^2 = ab and the greatest common divisor of a and b is equal to 1, prove that a and b are perfect squares

Homework Equations

I know that if (a,b)=1, then there exists integers u and v where 1=au+bv (even though i don't think this is necessary in this proof)

also, I know that the square root of a perfect square is a rational number, if it is not a perfect square, then it is irrational

Lastly, I know that since (a,b)=1 that means a and b are relatively prime

The Attempt at a Solution

I have absolutely no idea how to do this proof. I know i need to show that the square root of a and the square root of b are rational, but I don't know how to do that.

Maybe I could do it by trying to show it is irrational and finding a contradiction? Any help would be great!

 

[Dick]

It's pretty easy conceptually if you think about the prime factorizations of a and b. Try that.

 

[dancergirlie]

Yeah, I figured it out like 10 minutes after i posted, it is really easy now that I thought of the prime factorizations of a and b. Thanks for the help though!