Solution of Linear Diophantine Equation

Theorem
The linear Diophantine equation:
 * $$ax + by = c$$

has solutions iff $$\gcd \left\{{a, b}\right\} \backslash c$$ (that is, iff $$\gcd \left\{{a, b}\right\}$$ divides $$c$$).

If this condition holds with $$\gcd \left\{{a, b}\right\} > 1$$ then division by $$\gcd \left\{{a, b}\right\}$$ reduces the equation to:
 * $$a' x + b' y = c'$$

where $$\gcd \left\{{a', b'}\right\} = 1$$.

If $$x_0, y_0$$ is one solution of the latter equation, then the general solution is:
 * $$\forall k \in \Z: x = x_0 + b' k, y = y_0 - a' k$$.

Proof
We assume that both $$a$$ and $$b$$ are non-zero, otherwise the solution is trivial.

The first part of the problem is a direct restatement of Integer Combinations Multiples of GCD:

The set of all integer combinations of $$a$$ and $$b$$ is precisely the set of all integer multiples of the GCD of $$a$$ and $$b$$:


 * $$\gcd \left\{{a, b}\right\} \backslash c \iff \exists x, y \in \Z: c = x a + y b$$

Now, suppose that $$x', y'$$ is any solution of the equation.

Then we have:
 * $$a' x_0 + b' y_0 = c'$$ and $$a' x + b' y = c'$$.

Substituting for $$c'$$ and rearranging:
 * $$a' \left({x' - x_0}\right) = b' \left({y_0 - y'}\right)$$.

So $$a'$$ divides $$b' \left({y_0 - y'}\right)$$.

Since $$\gcd \left\{{a', b'}\right\} = 1$$, from Euclid's Lemma we have $$a'$$ divides $$y_0 - y'$$.

So $$y_0 - y' = a' k$$ for some $$k \in \Z$$.

Substituting into the above gives $$x' - x_0 = b' k$$ and so:
 * $$x' = x_0 + b' k, y' = y_0 - a'k$$ for some $$k \in \Z$$

which is what we claimed.

Substitution again gives that the integers:
 * $$x_0 + b' k, y_0 - a' k$$

constitute a solution of $$a' x + b' y = c'$$ for any $$k \in \Z$$.