Solution of Linear Diophantine Equation

Theorem
The linear Diophantine equation:
 * $a x + b y = c$

has solutions :
 * $\gcd \set {a, b} \divides c$

where $\divides$ denotes divisibility.

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

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

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

or:
 * $\forall k \in \Z: x = x_0 + \dfrac b d k, y = y_0 - \dfrac a d k$

where $d = \gcd \set {a, b}$.

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 Set of Integer Combinations equals Set of Multiples of GCD:

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


 * $\gcd \set {a, b} \divides c \iff \exists x, y \in \Z: c = x a + y b$

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:
 * $a' \paren {x' - x_0} = b' \paren {y_0 - y'}$

So:
 * $a' \divides b' \paren {y_0 - y'}$

Since $\gcd \set {a', b'} = 1$, from Euclid's Lemma we have:
 * $a' \divides \paren {y_0 - y'}$.

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 $k \in \Z$.