If gcd(a,c)=1 and b|c, then gcd(a,b)=1?

Question

Prove if $\gcd(a,c)=1$ and $b|c$, then $\gcd(a,b)=1$

Answer 1

$gcd(a,c)=1$ and $b|c$

$$\exists x, y \in \mathbb{Z}, ax+cy=1$$ and since we can write $c=bk$.

we can write $$ax+b(ky)=1$$

Hence $$gcd(a,b)=1$$

Answer 2

Write $d=(a,b)$
By definition, $d\mid a$ and $d\mid b$
By hypothesis, $d\mid c$
By definition again, $d\mid(a,c)=1$
Hence $d=1.$