Suppose $R$ is an integral domain and $a,b\in R$ have no common factors other than units. Then $1$ is a greatest common divisor of $a$ and $b$.
Is $1$ the only greatest common divisor, or can there be others?
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The GCD is very rarely unique. If $x$ is a GCD of $a$ and $b$, then so is any associate of $x$, that is, any number of the form $ux$ with $u$ a unit. This is easily verifiable from the definition
$gcd(a,b)=x$ if $x$ is a common divisor of $a$ and $b$ and is divisible by all other common divisors of $a$ and $b$
Thus the GCD is only unique when you have a ring with one unit. In your particular case, every unit is a GCD.