Linear Diophantine Equation/Examples/35x - 256y = 48

Example of Linear Diophantine Equation
The linear diophantine equation:
 * $35 x - 256 y = 48$

has the general solution:
 * $\tuple {x, y} = \tuple {-5616 + 256 t, -768 - 35 t}$

Proof
We use Solution of Linear Diophantine Equation.

Using the Euclidean Algorithm:

Hence we see that $\gcd \set {35, -256} = 1$ which trivially divides $48$, and so there exists a solution.

Again with the Euclidean Algorithm:

From Solution of Linear Diophantine Equation, the general solution is:


 * $\forall t \in \Z: n = -5616 + 256 t, k = -768 - 35 t$