The question, as posed doesn't make sense. Coprime means that two integers don't have a common divisor. gcd(a,b) is one integer-which is the other one? 

If a and b are coprime, which means that gcd(a,b)=1, then it's trivial that it divides any integer c. 

If gcd(a,b) does divide c, without gcd(a,b) being equal to 1, then this means that the three integers a, b and c have a common factor, that can be divided out.

If gcd(a,b) does not divide c, then it's clear that the LHS can never be equal to the RHS in integers. 

(Incidentally, the equation at hand doesn't have anything to do with RSA encryption.)

I concur with Mr. Stam, because if the gcd (a,b) is coprime, then the linear diophantine equation will always have an integer solution. But if the gcd (a,b)is not coprime, then there might be some possibilities such that the integer solution does not exist.

By comparing with RSA key equation: ed+kphi(n)=1 where the gcd (e,phi(n))=1.

Once more ``gcd (a,b) is coprime'' doesn't make sense and anything following from such a statement is, also, meaningless.

How the RSA algorithm works is described in many places, e.g. https://engineering.purdue.edu/kak/compsec/NewLectures/Lecture12.pdf

The answer given by Mr. Stoica really captured what I am looking for.