Prove that if integers $a ≠ 0$ and $b$ such that $a \mid b$ then $\gcd (a, b) = |a|$.
Since $a \mid b$, then for some integer $k$ we have
$$ b = ak $$
Substituting
$$ \begin{align} \gcd(a,b) & = \gcd(a, ak) \\ \\ &= \lvert a \rvert \gcd (\frac{a}{a},\frac{ak}{a}) \\ \\ &= \lvert a \rvert \end{align}$$
The penultimate line uses Proposition 1.5 that if $\gcd (a, b) = g$ then $\gcd (\frac{a}{g}, \frac{b}{g})=1$. We make use of $a \neq 0$ to ensure the division is valid.
The multiplier is $\lvert a \rvert$ because by definition a gcd is greater than zero.