# Proving Perfect Squares: A Study in Number Theory

• dancergirlie
In summary, if C^2 = ab and the greatest common divisor of a and b is equal to 1, it can be proven that a and b are perfect squares. This can be done by showing that the square root of a and the square root of b are rational, which can be achieved by considering their prime factorizations.
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!

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

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!

## What is a perfect square?

A perfect square is a number that is obtained when an integer is multiplied by itself. For example, 9 is a perfect square because it is equal to 3 x 3.

## How do you prove that a number is a perfect square?

To prove that a number is a perfect square, you can take its square root. If the square root is an integer, then the number is a perfect square. For example, the square root of 25 is 5, so 25 is a perfect square.

## What is the perfect squares theorem?

The perfect squares theorem states that every positive integer can be expressed as the square of another integer. In other words, every number has a square root.

## Is zero a perfect square?

Yes, zero is a perfect square because it can be expressed as 0 x 0.

## Can a negative number be a perfect square?

No, a negative number cannot be a perfect square because when we take the square root of a negative number, the result is a complex number, not an integer.

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