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?
