Given that $a$ and $b$ are coprime and $c>0$, I need to show that there exist some $x$ such that $a+xb$ and $c$ are coprime (everything here are integers).
My attempt was to write out $\alpha a + \beta b =1$ for some $\alpha,\beta$ and $\gamma b + \lambda c =g$ where $g=gcd(b,c)$, for some $\gamma,\lambda$. Multiplying both equations, I got $\alpha(a+bx)+Bc=1$, with $B=\beta \lambda b /g$ and $x=\beta \gamma b/\alpha g$. Since $b/g$ is an integer, this is very close to what I need. However, I am not sure that that $x$ is an integer here.
Let $x$ be the product of all the primes dividing $c$ but not $a$ (with $x=1$ if no such primes exist).
Suppose that $p$ is a prime dividing $c$. If $p|a$ then $p$ doesn't divide $x$ or $b$ (since $a$ and $b$ are coprime), so $p$ cannot divide $a+xb$. If $p$ doesn't divide $a$ then it divides $x$, so cannot divide $a+xb$.
Therefore no prime dividing $c$ can divide $a+xb$, so $c$ and $a+xb$ are coprime.