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. This means that when you multiply a by x and divide by m, the remainder is 1.

2. How Does the Calculator Work?

The calculator finds x such that:

Where:

• \( a \) — The integer whose inverse we want to find
• \( m \) — The modulus
• \( x \) — The modular inverse (result)

Explanation: The calculator uses a brute-force method to test each possible value of x until it finds one that satisfies the equation or determines that no inverse exists.

3. Importance of Modular Inverse

Details: Modular inverses are crucial in cryptography (especially RSA algorithm), computer algebra systems, and solving linear congruences. They're fundamental in number theory and have practical applications in secure communications.

4. Using the Calculator

Tips: Enter positive integers for both a and m. The inverse exists only when a and m are coprime (gcd(a,m) = 1). For large numbers, the calculation may take longer.

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: Is the modular inverse unique?
A: The inverse is unique modulo m, meaning 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 our calculator uses positive integers. For negative a, you can add m until you get a positive equivalent.

Q5: What are some practical applications?
A: Used in cryptography, error-correcting codes, and algorithms that require division in modular arithmetic.