1. What is Modular Inverse?

The modular inverse of a number a modulo m is the integer x such that \( a \times x \equiv 1 \mod m \). It exists if and only if a and m are coprime (gcd(a,m) = 1).

2. How Does the Calculator Work?

The calculator uses a brute-force approach to solve the equation:

Where:

• \( a \) — The integer whose inverse is sought
• \( m \) — The modulus
• \( x \) — The modular inverse (solution)

Explanation: The calculator tests each integer x from 1 to m-1 until it finds one that satisfies the equation.

3. Applications of Modular Inverse

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

4. Using the Calculator

Tips: Enter positive integers for both a and m. The calculator will find x if it exists, 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: Is the modular inverse unique?
A: Yes, modulo m. All solutions are congruent modulo m.

Q3: What's a more efficient algorithm than brute force?
A: The Extended Euclidean Algorithm is much more efficient, especially for large numbers.

Q4: Can negative numbers have modular inverses?
A: Yes, but this calculator uses positive integers. For negative a, use a ≡ a mod m.

Q5: What if m is 1?
A: The inverse is always 0 since any number × 0 ≡ 0 ≡ 1 mod 1.