Using extended Euklid to find a linear combination

Question

I got the following equation where you have to find possible integers for $m$ and $n$ that satisfy:

$$144n+136m=4$$

What I tried was to use the Extended Euklidean Algorithm. But it gives me the greatest common divisor which is 8 for the following combination:

$144*1+136*1=8$ where $m=1$ and $n=1$

Is there any way to find a linear combination of $m$ and $n$ that satisfies the equation above?

Or, is there no solution for this at all?

Thank you

score 2 · Accepted Answer · answered Feb 22 '15 at 18:54

2

No, you can't. For any integers $a,b$, the set $\bigl\{ua+vb\;\vert\:u,v\in\mathbf Z\bigr\}$ is the set of multiples of $\gcd(a,b)$. As this g.c.d. is $8$, you only can obtain multiples of $8$.

answered Feb 22 '15 at 18:54

Bernard

175,478

score 0 · Answer 2 · edited Apr 13 '17 at 12:21

0

Hint $\ $ Cancelling $\,4\,$ yields $\, 26n+34m = 1.\,$ LHS is even, contra RHS is odd.

Remark $\ $ Generally $\ a\Bbb Z\! +\! b\Bbb Z\overset{\rm Bezout} = \gcd(a,b) \Bbb Z\,$ so $\,ax\!+\!by = c\,$ is solvable $\iff \gcd(a,b)\mid c$

edited Apr 13 '17 at 12:21

Community

1

answered Feb 22 '15 at 18:59

Bill Dubuque

272,048

Ok, so its not possible to find a combination? – Ovomaltine Feb 22 '15 at 19:14
@Ovomaltine Right, since if a solution $,m,n,$ exists then, by the my first equation above, there is an integer that is both even and odd, a contradiction. – Bill Dubuque Feb 22 '15 at 19:23

Using extended Euklid to find a linear combination

2 Answers2