# Linear diophantine equation

a) $$130x + 143y = 5957$$
b) $$44x + 19y = 75$$

the theorem says ax + by = c has solution if and only if d | c

however I work out both question with no solution as

a)

$$143 = 130 . 1 + 13$$
$$130 = 13 . 10 + 0$$

and 5957 is unable to be divided by 13 therefore no solution exists.

b)

$$44 = 19 . 2 + 6$$
$$19 = 6 . 3 + 1$$
$$6 = 3 . 2 + 0$$

for b) I am unsure if there are any solution exists?

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Can you try to clean up your question please? It is hard for me to tell exactly what you are doing and what you are asking, without simply assuming that you are trying to apply Bezout's identity and getting stuck somewhere. –  Alex Becker May 1 '11 at 6:31

Your second one, however, is not. When doing the Euclidean Algorithm, you mess up between your second and third steps. From $19 = 6*3 + 1$, you should go on the next next step $6 = 1*6 + 0$, indicating that 1 is the gcd of 19 and 44.
@Liangteh: I will not do your homework for you, but I will do a quick example. Consider the numbers 3 and 8. $8 = 3*2 + 2$, and $3 = 2 * 1 + 1$. So 1 is the gcd. But then, by reversing my equations, I get that $1 = 3 - 2(1) = 3 - (8 - 3(2)) = 3*(3) - 8*(1)$. So I have a solution. –  mixedmath May 1 '11 at 6:52