Linear Diophantine Equation

Theorem.

Let with a, not both zeros then LDE

has a solution over
Also if is a solution af the LDE then every other soution has the form

where and

Proof.
If LDE has a solution then clearly

suppose
then we can write

for some then

and
let be a given solution
let be any other solution then

then

□

Find all solution of

solution we apply the euclidian algorithm to 7 and 19 we can do it

method 2

Let be the non-zero column and the zero column
and value of
then