Definition:Greatest Common Divisor

Integers
When the integral domain in question is the integers $\Z$, the GCD is usually defined differently, as follows:

Real Numbers
The concept can be extended to the set of real numbers:

Equivalence of Definitions
In GCD iff Divisible by Common Divisor it is demonstrated that the definition of GCD as specified for the integers is logically equivalent to the definition as given for the GCD as specified for an integral domain.

Note, however, that an integral domain is not an ordered set and so the definition as given for integers can not apply, as there is then no concept of greatest.

The definition for the integral domain is an abstraction from the definition as initially encountered in grade-school arithmetic, and many treatments of number theory and abstract algebra start with the definition as given for an integral domain on which the ordering $\le$ is then applied.

When considering an ordered integral domain, it is of course possible to use either definition. It is as well to make sure which one is meant.

Also known as
The greatest common divisor is also known as the highest common factor (abbreviated HCF or hcf) and written $\operatorname{hcf} \left\{{a, b}\right\}$ or $\operatorname{hcf} \left({a, b}\right)$.

Alternatively, $\gcd \left\{{a, b}\right\}$ is written in some texts as $\left({a, b}\right)$, but this notation can cause confusion with ordered pairs.

The notation $\gcd \left({a, b}\right)$ is also seen, but the set notation, although arguably more cumbersome, can be argued to be preferable.

The archaic term greatest common measure can also be found, mainly in such as Euclid's.

Also see

 * Elements of Euclidean Domain have Greatest Common Divisor where it is shown that any two GCDs of $a$ and $b$ are associates.