Why do we notate the greatest common divisor of $a$ and $b$ as $(a,b)$?

Solution 1:

The pair / tuple notation used both for gcds and ideals serves to highlight their similarity. Just as in the domain $\,\Bbb Z,\,$ in any PID we have the ideal equality $\,(a,b) = (c)\iff \gcd(a,b) \cong c,\,$ where the congruence means "associate", i.e. they divide each other (differ by only a unit factor). Thus in a PID we can equivalently view $\,(a,b)\,$ as denoting either a gcd or an ideal, and the freedom to move back-and-forth between these viewpoints often proves useful.

Gcds and ideals share many properties, e.g. associative, commutative, distributive laws, and

$$ b\equiv b'\!\!\!\pmod{\!a}\,\Rightarrow\, (a,b) = (a,b')$$

Using the shared properties and notation we can give unified proofs of theorems that hold true for both gcds and ideals, e.g. in the proofs below we can read the tuples either as gcds or ideals

$$(a,b)\,(a^2,b^2)\, =\, (a,b)^3\ \ \ {\rm so}\ \ \ (a,b)=1\,\Rightarrow\, (a^2,b^2) = 1$$

$\quad \color{#c00}{ab = cd}\ \Rightarrow\ (a,c)\,(a,d)\, =\ (aa,\color{#c00}{cd},ac,ad)\, =\, \color{#c00}a\,(a,\color{#c00}b,c,d)\,\ [= (a)\ \ {\rm if}\ \ (a,c,d) = 1] $

Such abstraction aids understanding generalizations and analogies in more general ring-theoretic contexts - which will become clearer when one studies divisor theory, e.g. see the following

Friedemann Lucius. Rings with a theory of greatest common divisors.
manuscripta math. 95, 117-36 (1998).

Olaf Neumann. Was sollen und was sind Divisoren?
(What are divisors and what are they good for?) Math. Semesterber, 48, 2, 139-192 (2001).