1. What is Modular Inverse?

The modular 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, a and x are inverses of each other under multiplication modulo m.

2. How Does the Calculator Work?

The calculator uses the following mathematical relationship:

Where:

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

Explanation: The calculator finds the smallest positive integer x that satisfies the equation (a × x) mod m = 1.

3. Importance of Modular Inverse

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

4. Using the Calculator

Tips: Enter positive integers for both a and m. Note that a modular inverse only exists when a and m are coprime (gcd(a,m) = 1).

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 (their greatest common divisor is 1).

Q2: What's the time complexity of finding modular inverse?
A: The naive approach (used here) is O(m). The Extended Euclidean Algorithm can find it in O(log(min(a,m))) time.

Q3: Can zero have a modular inverse?
A: No, zero never has a modular inverse since 0 × x ≡ 0 mod m for any x.

Q4: Are modular inverses unique?
A: Yes, modulo m. If x is an inverse, then x + km for any integer k is also an inverse, but we typically use the smallest positive one.

Q5: What are some practical applications?
A: Used in cryptography, error detection/correction codes, random number generation, and solving congruences.