Gcd(a,b) unique in Euclidean domain?

[julypraise]

gcd(a,b) unique in Euclidean domain??

Homework Statement

In Hungerford's Algebra on page 142, the problem 13 describes Euclidean algorithm on a Euclidean domain R to find THE greatest common divisor of a,b in R.

My question is that does this THE mean THE UNIUQE? I've heard from my lecturer in a general commutative ring, a greatest common divisor of a,b in R does not have to be unique.
Is there any theorem such as states that if R is a Euclidean domain, then for any a,b in R, gcd(a,b) is unique?

Homework Equations

The Attempt at a Solution

Sorry, I have not figured out at all...

 

[Hurkyl]

It's not even unique in the integers: 5 and -5 are both greatest common divisors of 20 and 35, for example. However, there is a simple relationship between all of the possibilities for a gcd...