Let we have the following form :

ax+by=c, a,b,c are integers. Further I am interested in finding all the integral solution of this equation. This equation is called linear Diopantine equation in two variables.

Today I will prove that this equations has a solution if and only if the greates common divisor of a and b, divides c.Let d=gcd(a,b) and let x' and y' be a solution, then d|ax'+by' , which means that d|c. Now let d|c, then there is an integer t such that c=dt, but there is integers u and v such that d=au+bv, then c=(au+bv)t, which means that c=a(ut)+b(vt), thus ut,vt is a solution, and the proof is complete. Later I will formulate the general form of the solution.