Since the gcd of two numbers is not unique, shouldn't "gcd" be read "great common divisor" instead of "greatest common divisor"?

Question

I take it that a gcd of $a$ and $b$ is, by definition, a common divisor of $a$ and $b$, and a multiple of every common divisor of those two elements.

And, if I am not mistaken, more than one element can satisfy these two criteria.

Therefore, "gcd" should, I think, be read "great common divisor" and not "greatest common divisor".

Do you agree?

Answer 1

Two different greatest common divisors can only differ in the sign, so you can talk of positive and negative greatest common divisor. The g part of gcd means that $\gcd$ is a maximum in the set of lower bounds of two (or more) elements, where the order relation on the set of elements is given by divisibility, which on the integers is not antisymmetric.

Answer 2

The defense of using 'greatest' is that gcd, when considered on ideals, is unique. It is defined as follows: $K = gcd(I,J)$ if and only if $K\supset I, K\supset J$ and $K$ is the largest such ideal w.r.t. containment. When ideals are taken to be principal (i.e. generated by one element), this definition gives 'usual' gcd.