Solve and explain diophantine equation

Question

1. A Diophantine equation ax+by = c always has a solution whenever a and b are relatively prime.
2. Find x ,y such that $$93x-81y=3 $$

Answer 1

Divide throughout by $3$. You get $31x-27y=1$. Now note that $gcd(31,27)=1$ Thus, by Euclidean algorythm there exist integers $p$ and $q$ such that. $31p+27q=1$. Compare to get values of $x$ and $y$.

You have,

$31=27(1)+4$

$27=4(6)+3$

$4=3(1)+1$

Thus you have

$1=4-3$

$=4-(27-4(6))$

$=4(7)-27$

$=7(31-27)-27$

$=31(7)-27(8)$.

Thus $31(7)+27(-8)=1$

Answer 2

Infinitely many solutions of the same.

$y=31k+8$

$x=27k+7$

or

$y=31k-23$

$x=27k-20$