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# Linear Diophantine equations

This calculator solves linear diophantine equations.

As usual, here goes the calculator, and theory goes below it.

### Linear Diophantine equations

Equation

All solutions for x

All solutions for y

x

y

Since this is all about math, I copy some content from wikipedia for the start.

In mathematics, a Diophantine equation is a polynomial equation in two or more unknowns such that only the integer solutions are searched or studied (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.

The simplest linear Diophantine equation takes the form

$ax + by = c$,

where a, b and c are given integers, x, y — unknowns.

The solutions are completely described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if (x, y) is a solution, then the other solutions have the form (x + kv, y - ku), where k is an arbitrary integer, and u and v are the quotients of a and b (respectively) by the greatest common divisor of a and b.

To find the solution one can use Extended Euclidean algorithm (except for a = b = 0 where either there are unlimited number of solutions or none).

If a and b are positive integers, we can find their GCD g using Extended Euclidean algorithm, along with $x_g$ и $y_g$, so:

$ax_g + by_g = g$.

If c is a multiple of g, the Diophantine equation $ax + by = c$ has solution, otherwise, there is no solution.

That is, if c is a multiple of g, then

$a x_g (\frac{c}{g}) + b y_g (\frac{c}{g})=c$

and one of the possible solutions is:

$x_0 = x_g (\frac{c}{g})$$y_0 = y_g(\frac{c}{g})$

If either a or b is negative, we can solve equation using it's modulus, then change sign accordingly.

If we know one of the solutions, we can find their general form.

Let g = GCD(a,b), and we have:

$ax_0 + by_0 = c$.

By adding $\frac{b}{g}$ to $x_0$ and subtracting $\frac{a}{g}$ from $y_0$, we get:

$a(x_0 + \frac{b}{g}) + b(y_0 - \frac{a}{g}) = ax_0+by_0 + \frac{ab}{g}-\frac{ba}{g}=c$

So, any numbers like these:
$x = x_0 + k \frac{b}{g}$$y = y_0 - k \frac{a}{g}$,

where k is integer, are the solutions of Linear Diophantine equation.