Solve Ax ≡ B mod C w/ Bezout's Identity

[TMO]

This is a very obvious question, but I am having trouble concentrating. Let ax ≡ b mod c and let gcd(a, c) | b. How do I convert this equation into Bezout's identity so that I can use the extended Euclidean algorithm?

 

[fresh_42]

This is a very obvious question, but I am having trouble concentrating. Let ax ≡ b mod c and let gcd(a, c) | b. How do I convert this equation into Bezout's identity so that I can use the extended Euclidean algorithm?

You can divide by $\operatorname{gcd}(a,b)$ all three numbers to get $a'x \equiv b' \operatorname{mod} c'$ with now $\operatorname{gcd}(a',b')=1$ and use the Euclidean division to find $1=na'+mb'$.

But somehow I'm not sure whether you meant this or something else.