0
$\begingroup$

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?

$\endgroup$
  • 2
    $\begingroup$ $95-13n = 4-13(n-7)$, $-152+21n = -5+21(n-7)$. $\endgroup$ – Abstraction Mar 17 '17 at 21:46
  • $\begingroup$ @Abstraction Ah...I missed that. Thanks. $\endgroup$ – TheGeekGreek Mar 17 '17 at 21:48
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.