Linear Diophantine Equation Solver
Solve linear Diophantine equations ax + by = c and find all integer solutions
About this calculator
The Linear Diophantine Equation Solver finds all integer solutions to equations of the form ax + by = c, where a, b, and c are given integers. These equations appear frequently in number theory, cryptography, computer science, and mathematical competitions. The calculator determines whether solutions exist using the greatest common divisor condition, then provides the general solution formula when integer solutions are possible, making it invaluable for students and professionals working with modular arithmetic and discrete mathematics.
How to use
Enter the coefficients a, b, and c for your linear Diophantine equation ax + by = c into the respective input fields. Click 'Solve' to calculate the results. The calculator will determine if integer solutions exist and display either the general solution formula or indicate that no integer solutions are possible.
Frequently asked questions
When does a linear Diophantine equation have integer solutions?
A linear Diophantine equation ax + by = c has integer solutions if and only if gcd(a,b) divides c evenly.
How many solutions does a linear Diophantine equation have?
If solutions exist, there are infinitely many integer solutions following a specific parametric pattern involving one free parameter.
What is the extended Euclidean algorithm's role in solving these equations?
The extended Euclidean algorithm finds initial particular solutions by expressing gcd(a,b) as a linear combination of a and b.