Divisibility theory (Z)/Lemma of Bezout/Fact

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

has a greatest common divisor

{}d

, and this

{}d

can be expressed as a linear combination of the

{}a_{1},\ldots ,a_{n}

, that is, there exist integers

{}r_{1},\ldots ,r_{n}

such that

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

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