How do you refer to this Term in English?

Question

How do you refer to this theorem in English exactly?

$$a\mathbb Z+ b\mathbb Z =d\mathbb Z \text{, where }d = \gcd(a,b) \text{ and a, b}\in \mathbb Z$$

I imagine it should be something like: "The set of all integer linear combinations is..."

Regards

Answer 1

If restricted to elementary number theory (as tagged). I would write "the set of all integral linear combinations of $\,a,b\,$ equals the set of all integral multiples of their gcd", whose element-wise form is widely referred to as Bezout's identity for the gcd.

More elementarily, I would say that any common divisor $\,c\,$ of $\,a,b\,$ of linear form $\,c = ja+kb,\,$ $\,j,k\in\Bbb Z,\,$ is necessarily a greatest common divisor, since $\,d\mid a,b\,\Rightarrow\, d\mid ja+kb = c,\,$ so $\,d\le c.$

Answer 2

It's called Bézout's identity.

The most elementary way of saying it is to use the phrase integer combination: "the set of all integer combinations of $a$ and $b$ is the set of multiples of $\gcd(a, b)$".