Linear Diophantine Equation/Examples/121x - 88y = 572

Example of Linear Diophantine Equation
The linear diophantine equation:
 * $121 x - 88 y = 572$

has the general solution:
 * $\tuple {x, y} = \tuple {156 - 8 t, 208 - 11 t}$

Proof
Using the Euclidean Algorithm:

Thus we have that:
 * $\gcd \set {121, -88} = 11$

which is a divisor of $572$:
 * $572 = 52 \times 11$

So, from Solution of Linear Diophantine Equation, a solution exists.

Next we find a single solution to $121 x - 88 y = 572$.

Again with the Euclidean Algorithm:

and so:

is a solution.

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


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

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

In this case:

giving: