Multiplicative Inverse Modulo Calculator
Result
About Multiplicative Inverse Modulo
The modular multiplicative inverse of a number a modulo m is an integer x such that (a × x) mod m = 1. It exists only if a and m are coprime (their greatest common divisor is 1).
Formula:
a × x ≡ 1 (mod m), where gcd(a, m) = 1
Properties:
- The inverse exists only if a and m are coprime
- The inverse is unique modulo m
- If a and m are not coprime, no inverse exists
- The inverse is always between 0 and m-1
Examples:
| Number (a) | Modulus (m) | Inverse (x) | Verification |
|---|---|---|---|
| 3 | 11 | 4 | 3 × 4 ≡ 1 (mod 11) |
| 7 | 13 | 2 | 7 × 2 ≡ 1 (mod 13) |
| 5 | 12 | 5 | 5 × 5 ≡ 1 (mod 12) |
Applications:
- Cryptography (RSA algorithm)
- Solving linear congruences
- Error detection and correction
- Number theory problems
- Computer security
Tips:
- Always check if the numbers are coprime first
- The inverse is always less than the modulus
- For prime modulus, all non-zero numbers have inverses
- Use the Extended Euclidean Algorithm for large numbers
- Remember that the inverse is unique modulo m
Advertisement Space