The discussion focuses on solving the modular equation ax ≡ b mod c using Bezout's Identity and the extended Euclidean algorithm. It is established that if gcd(a, c) divides b, the equation can be transformed by dividing all terms by gcd(a, b) to yield a' x ≡ b' mod c' where gcd(a', b') = 1. This transformation allows for the application of the extended Euclidean algorithm to find integer solutions.
PREREQUISITESMathematicians, computer scientists, and students studying number theory or cryptography who need to solve linear congruences and understand modular equations.
You can divide by ##\operatorname{gcd}(a,b)## all three numbers to get ##a'x \equiv b' \operatorname{mod} c'## with now ##\operatorname{gcd}(a',b')=1## and use the Euclidean division to find ##1=na'+mb'##.TMO said:This is a very obvious question, but I am having trouble concentrating. Let ax ≡ b mod c and let gcd(a, c) | b. How do I convert this equation into Bezout's identity so that I can use the extended Euclidean algorithm?