If an integer $a$ is coprime with an integer $b$, then will the integer $a$ also be coprime with the integer $9a + b$?

Question

If an integer $a$ is coprime with an integer $b$, then will the integer $a$ also be coprime with the integer $9a + b$ ?

Answer 1

If $1=ma+nb$, then $1=(m-9n)a+n(9a+b)$.

Answer 2

Hint $\ $ The linear map $\,(a,b)\to (a,9a\!+\!b)\,$ has det $=\color{#c00}1,\ $ hence $\ \gcd(a,9a\!+\!b)\mid \color{#c00}1\cdot \gcd(a,b)$