1. What is Modular Inverse?

The modular inverse of a number a modulo m is an integer x such that the product a × x is congruent to 1 modulo m. It exists if and only if a and m are coprime (gcd(a, m) = 1).

2. How Does the Calculator Work?

The calculator finds x that satisfies:

Method: The calculator uses a brute-force approach to test each integer x from 1 to m-1 until it finds one that satisfies the equation.

3. Importance of Modular Inverse

Applications: Modular inverses are essential in cryptography (especially RSA algorithm), computer algebra systems, and solving linear congruences.

4. Using the Calculator

Tips: Enter integer values for a and m (both > 0). The calculator will find x if it exists, or indicate if no inverse exists (when a and m are not coprime).

5. Frequently Asked Questions (FAQ)

Q1: When does a modular inverse exist?
A: A modular inverse exists if and only if a and m are coprime (gcd(a, m) = 1).

Q2: What's the difference between this and division?
A: Modular inverse is the multiplicative inverse in modular arithmetic, not regular division.

Q3: Is the inverse unique?
A: The inverse is unique modulo m, meaning all solutions are congruent modulo m.

Q4: How is this used in cryptography?
A: In RSA, modular inverses are used to compute private keys from public keys.

Q5: What's a more efficient algorithm than brute-force?
A: The Extended Euclidean Algorithm is much more efficient, especially for large numbers.