# Finding integer solutions of one equation with two unknowns

Consider the linear system $$357 x + 221 y = 323$$

We are looking for the integer solutions, i.e. solutions of the form $(x,y) \in \mathbb{Z}^2$. There is a way of finding a particular solution using the euclidean algorithm and then adding integer multiples of a certain structure. We get $$(x,y) = (95-13n,-152+21n) \qquad n \in \mathbb{Z}$$ The lecturer gave also the equivalent set of solutions $$(x,y) = (4 - 13m, -5 + 21m) \qquad m \in \mathbb{Z}$$ However I do not quite see how one gets the equivalent formulation. Furthermore I am asked to find the minimal solution $(x,y)$, i.e. the solution where $|x| + |y|$ is minimal. Has anyone a hint for finding this?

• $95-13n = 4-13(n-7)$, $-152+21n = -5+21(n-7)$. – Abstraction Mar 17 '17 at 21:46
• @Abstraction Ah...I missed that. Thanks. – TheGeekGreek Mar 17 '17 at 21:48

Because if there is integers satisfying $357x+221y = 323$ then it must also be satisfying $\frac{357}{17}x+\frac{221}{17}y=\frac{323}{17}$ which is just $21x+13y=19$ if you solve this using the euclidean algorithm and then adding integer multiples of a certain structure you will get the desired solutions.