Linear Diophantine Equation/Examples/23x + 29y = 25

Example of Linear Diophantine Equation
The linear diophantine equation:
 * $23 x + 29 y = 25$

has the general solution:
 * $\tuple {x, y} = \tuple {-125 + 29 t, 100 - 23 t}$

Proof
Using the Euclidean Algorithm:

Thus we have that:
 * $\gcd \set {23, 29} = 1$

which is (trivially) a divisor of $25$.

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

Next we find a single solution to $23 x + 29 y = 25$.

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: