Divisibility theory (Z)/Lemma of Bezout/Fact

Every set of integers ${\displaystyle {}a_{1},\ldots ,a_{n}}$

has a greatest common divisor ${\displaystyle {}d}$, and this ${\displaystyle {}d}$ can be expressed as a linear combination of the ${\displaystyle {}a_{1},\ldots ,a_{n}}$, that is, there exist integers ${\displaystyle {}r_{1},\ldots ,r_{n}}$ such that

${\displaystyle {}r_{1}a_{1}+r_{2}a_{2}+\cdots +r_{n}a_{n}=d\,.}$

In particular, for coprime integers ${\displaystyle {}a_{1},\ldots ,a_{n}}$, there exists a representation of ${\displaystyle {}1}$.