# Solve Ax ≡ B mod C w/ Bezout's Identity

• B
• TMO
In summary, the question is about converting an equation ax ≡ b mod c into Bezout's identity to use the extended Euclidean algorithm. The solution involves dividing all three numbers by the greatest common divisor of a and c and using Euclidean division to find a solution for the equation.
TMO
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?

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?
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'##.

But somehow I'm not sure whether you meant this or something else.

## 1. What is the purpose of using Bezout's Identity to solve Ax ≡ B mod C?

Bezout's Identity is a mathematical theorem that allows us to find the greatest common divisor (GCD) of two integers. In the context of solving Ax ≡ B mod C, it helps us find a solution to the congruence equation by determining the inverse of A modulo C.

## 2. How does Bezout's Identity work?

Bezout's Identity states that for any two integers A and B, there exist two other integers, x and y, such that Ax + By = GCD(A, B). This means that we can use the extended Euclidean algorithm to find the GCD and the values of x and y, which in turn can help us find the inverse of A modulo C.

## 3. Can Bezout's Identity be used to solve any congruence equations?

No, Bezout's Identity can only be used for solving linear congruence equations of the form Ax ≡ B mod C, where A and C are coprime (i.e. they have no common factors other than 1). If A and C are not coprime, then the congruence equation may not have a solution.

## 4. What are the limitations of using Bezout's Identity to solve congruence equations?

One limitation is that it only works for linear congruence equations. It cannot be used for quadratic or higher degree congruence equations. Additionally, if the modulus C is very large, it may be computationally expensive to find the inverse of A modulo C using Bezout's Identity.

## 5. Are there any other methods for solving Ax ≡ B mod C?

Yes, there are other methods such as the Chinese Remainder Theorem and Fermat's Little Theorem. These methods may be more efficient for certain types of congruence equations, and they can also be used for higher degree equations. However, Bezout's Identity is still a valuable tool for solving linear congruence equations and can be used in conjunction with other methods for more complex problems.

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