1. What is Modular Multiplicative Inverse?

The modular multiplicative inverse of an integer a modulo m is an integer x such that the product a × x is congruent to 1 modulo m. In other words, it satisfies the equation a × x ≡ 1 mod m.

2. How Does the Calculator Work?

The calculator finds x that satisfies:

Explanation: The calculator uses a brute-force method to test each possible value of x from 1 to m-1 until it finds one that satisfies the equation. If no such x exists (when a and m are not coprime), it returns "No inverse exists".

3. Importance of Modular Inverse

Details: Modular inverses are essential in cryptography (especially RSA algorithm), computer algebra systems, and solving linear congruences. They allow "division" in modular arithmetic.

4. Using the Calculator

Tips: Enter integer a and modulus m (must be > 1). The inverse exists only if a and m are coprime (gcd(a,m) = 1).

5. Frequently Asked Questions (FAQ)

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

Q2: What's the difference between inverse modulo m and regular division?
A: Modular inverse is the equivalent of reciprocal in modular arithmetic, allowing "division" by multiplication.

Q3: How is this used in cryptography?
A: RSA encryption uses modular inverses to compute private keys from public keys.

Q4: Is there a faster algorithm than brute-force?
A: Yes, the Extended Euclidean Algorithm is much more efficient, especially for large numbers.

Q5: Can zero have a modular inverse?
A: No, zero never has a modular inverse since gcd(0,m) = m ≠ 1 for m > 1.