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okay in this video we want to look at the following question given natural numbers a B and C when does there exist integers x and y such that ax + b y equals c so this is known as a linear Diophantine equation and were trying to solve a linear diet and Diet Fantini equation so lets recall the following if C equals the GCD of a and B then there does exist so there exists x and y integers such that ax + b y equals GCD of a and b and so finding such XY uses something called the extended Euclidean algorithm which I have a few videos on so and in fact theyre infinitely infinitely many solutions for this x and y and i give an example of that as well so what well focus on for the rest of this video is are there any other values of C that allow us to solve this equation and what are they so Ill clean up the chalkboard and then well get into that question okay good so now well get on to answering the question is when can we find a solution to ax + B y equals equals C where a B and C ar