About This Calculator

1. What is this calculator for?
2. Can I embed this on my website?
3. How can I solve a Diophantine equation manually?

What is this calculator for?

This calculator is designed to solve Diophantine equations of the form: \[ ax + by = c \] where \( a \), \( b \), and \( c \) are given integers, and the goal is to find integer solutions \( x \) and \( y \).

A Diophantine equation is a polynomial equation where only integer solutions are sought. The simplest form, and the one our calculator addresses, is a linear Diophantine equation. These equations are named after the ancient Greek mathematician Diophantus, who studied them extensively.

Can I embed this on my website?

Sure. Embedding is allowed as long as you promise to follow our conditions. Here's the embed code:

How can I solve a Diophantine equation manually?

To solve a Diophantine equation of the form \( ax + by = c \) manually, one common approach is to use the Extended Euclidean Algorithm. This method provides integer coefficients \( x \) and \( y \) such that: \[ ax + by = \gcd(a,b). \] If \( c \) is a multiple of \( \gcd(a,b) \), then the equation \( ax + by = c \) has solutions. If not, it doesn't.

The general procedure using the Extended Euclidean Algorithm is as follows:

1. Find the greatest common divisor of \( a \) and \( b \) using the Euclidean Algorithm.
2. Work backwards to express \( \gcd(a,b) \) as \( ax + by \).
3. If \( c \) is divisible by \( \gcd(a,b) \) (i.e., \( \gcd(a,b) | c \)), then the equation has solutions. Multiply both sides of the equation by \( \frac{c}{\gcd(a,b)} \) to find the solutions \( x \) and \( y \).