Multiplicative Inverse Modulo Calculator

Your details

Multiplicative: find x so a*x = 1 mod m. Additive: find x so a+x = 0 mod m (always exists, x = m - a mod m).
The integer whose modular inverse you want. Can be negative.
The modulus m. Must be 2 or greater.
Extended Euclidean works for all coprime pairs. Fermat's Little Theorem is a shortcut (a^(m-2) mod m) that only applies when m is prime.
Inverse (x)
4

The modular inverse in the range [0, m-1]

GCD(a, m)1
Verification3 × 4 ≡ 1 (mod 11)
Inverse exists?Yes - GCD(a, m) = 1
Inverse x4
GCD(a, m)1

Multiplicative inverse of 3 mod 11 is 4.

  • 3 times 4 equals 12, and 12 mod 11 = 1. The inverse is confirmed.
  • The modulus 11 is prime, so every integer from 1 to 10 has a multiplicative inverse mod 11.
  • The inverse is unique in the range [0, 10]. Adding any multiple of 11 gives another valid (but equivalent) solution.
  • Multiplicative inverses mod a prime are the foundation of RSA encryption and elliptic-curve cryptography.

Next stepUse this inverse to solve linear congruences: if a×x ≡ b (mod m), the solution is x ≡ b×a^(-1) (mod m).

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