A Linear Diophantine Equation is a type of equation that takes the form ax + by = c , where a , b , and c are integers, and x and y are unknown integers. The goal is to find integer solutions for x and y . These equations are named after the ancient Greek mathematician Diophantus, who studied equations that require integer solutions. To determine if a solution exists, the greatest common divisor (gcd) of a and b must divide c . If a solution exists, there are infinitely many solutions that can be expressed in a specific form. Linear Diophantine Equations are important in number theory and have applications in areas such as cryptography and computer science.