Q:

If $d$ is the integer which generates $a \mathbb Z + b \mathbb Z$

Michael Artin in Proposition 2.6 states:

Let $a,b$ be integers not both zero, and let $d$ be the positive integer which generates the subgroup $a \mathbb Z +b \mathbb Z$ then

$d$ can be written in the form of $ar+bs$
$d$ divides both $a,b$
If an integer $e$ divides $a$ and $b$, it also divides $d$

My questions:

How does the second proposition hold true for $a=2,b=7 , d=9$?
How do we prove this? (a slight hint may be sufficient)

A:

Question 1: It doesn't, but these three numbers don't satisfy the assumptions because $2\mathbb{Z}+7\mathbb{Z}$ isn't generated by $9$. (As pointed out slightly faster by Julian).
Question 2: This follows from the fact that $\{a,b\}\subset a\mathbb{Z}+b\mathbb{Z}$. (Here I have taken your second question to mean "how do we prove the second proposition?").

A:

$2\mathbb{Z}+7\mathbb{Z}=\mathbb{Z}$, not $9\mathbb{Z}$:
$2\cdot (-3)+7\cdot (1)=1$.

