I just started with number theory and encountered Euclid ' Algorithm to find greatest common divisor. Also I understand that greatest common divisor of two integers can be expressed as linear combination of two. I was just wondering if same could be done for greatest common divisor of three integers and I can prove that $ gcd(a,b,c) = ax +by +cz $ where $x,y,z$ are integers. Naturally arises the question how to find solutions of diophantine Equation :
$ax +by +cz = td$ where$ d$ is gcd of $a,b,c$ and $t$ is integer. Firstly it is difficult to find even a single solution to this and if in case by hit and trial I get a solution say $(x',y',z')$ then how can find all solution parametrized in terms of the known solution.