Linear Diophantine Equation/Examples/17x + 19y = 23

Example of Linear Diophantine Equation
The linear diophantine equation:
 * $17 x + 19 y = 23$

has the general solution:
 * $\tuple {x, y} = \tuple {207 + 19 t, -184 - 17 t}$

Proof
Using the Euclidean Algorithm:

Thus we have that:
 * $\gcd \set {17, 19} = 1$

which is (trivially) a divisor of $23$.

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

Next we find a single solution to $17 x + 19 y = 23$.

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: