1. What is Multiplicative Inverse Modulo?

The multiplicative inverse of a number a modulo m is a number x such that the product a × x is congruent to 1 modulo m. This inverse exists if and only if a and m are coprime (their greatest common divisor is 1).

2. How Does the Calculator Work?

The calculator uses the Extended Euclidean Algorithm to find integers x and y such that:

When gcd(a, m) = 1, the coefficient x is the modular inverse of a modulo m.

Algorithm Steps:

1. Apply the Extended Euclidean Algorithm to find gcd(a, m) and coefficients x, y
2. If gcd(a, m) ≠ 1, no inverse exists
3. Otherwise, adjust x to be positive (using modulo m) to get the inverse

3. Importance of Modular Inverse

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

4. Using the Calculator

Tips: Enter integer values for a and m (m must be > 1). The calculator will find x such that (a × x) mod m = 1, or indicate if no inverse exists.

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 additive and multiplicative inverse?
A: Additive inverse is about addition (x where a + x ≡ 0 mod m), while multiplicative inverse is about multiplication.

Q3: How is this used in cryptography?
A: In RSA, modular inverses are used to compute the private key from the public key.

Q4: Can zero have a modular inverse?
A: No, since gcd(0, m) = m ≠ 1 for m > 1.

Q5: What if m is prime?
A: When m is prime, every integer a not divisible by m has an inverse modulo m.