elementary number theory - Prove that if $gcd( a, b ) = 1$ then $gcd( ac, b ) = gcd( c, b ) $

I know it might be too easy for you guys here. I'm practicing some problems in the textbook, but this one really drove me crazy.
From $\gcd( a, b ) = 1$, I have $ax + by = 1$, where should I go from here? The extra $ac$ is so annoying. Any hint?

Thanks,
Chan

Answer

Setup: Let $d_1 = \gcd(c,b)$ and $d_2 = \gcd(ac,b)$.

So we have $cx_1 + by_1 = d_1$, $acx_2 + by_2 = d_2$, and $ax + by = 1$.

Step 1: Multiply $ax+by = 1$ by $d_1$ (using $d_1 := cx_1 + by_1$) and rearrange to show that $d_2|d_1$.

$$\begin{aligned}
d_1 (ax+by &= 1)\\

\implies ax(cx_1 + by_1) + bd_1y &= d_1 \\
\implies a c (x x_1) + b(a x y_1 + d_1 y) &= d_1
\end{aligned}$$

Since we know that $d_2 = \gcd(ac,b)$ divides any integer linear combination of $ac$ and $b$, we have $d_2 | d_1$.

Step 2: By a similar argument, multiply $ax+by = 1$ by $d_2$ (using $d_2 := acx_2 + by_2$) and rearrange to show that $d_1|d_2$.

$$\begin{aligned}
d_2 (ax+by &= 1) \\

\implies ax(acx_2 + by_2) + bd_2y &= d_2 \\
\implies c (a^2 x x_2) + b(a x y_2 + d_2 y) &= d_2
\end{aligned}$$

Since we know that $d_1 = \gcd(c,b)$ divides any integer linear combination of $c$ and $b$, we have $d_1 | d_2$.

Conclusion: Finally, because we have $d_1 | d_2$, $d_2 | d_1$, and $d_1$ and $d_2$ are non-negative (since they are the gcd of two integers), we conclude that $d_1 = d_2$. Thus, $gcd(ac,b)=gcd(c,b)$.