Multiplicative Inverse Modulo Calculator

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Modular Multiplicative Inverse Calculator

Find the inverse of a number modulo m and review the extended Euclidean steps beside the answer.

Last updated: March 2026 | By ForgeCalc Engineering

Modular Inverse Solver

RSA helper

Extended Euclidean Steps

1.Find x such that (3 × x) mod 11 = 1
2.r = 11, newr = 3, quotient = 3
3.r = 3, newr = 2, quotient = 1
4.r = 2, newr = 1, quotient = 2
5.Greatest common divisor: 1
6.Normalized inverse: 4
7.Check: (3 × 4) mod 11 = 1
Modular Inverse
4

(3 × 4) mod 11 = 1

What Is a Modular Multiplicative Inverse?

A modular multiplicative inverse is the number that multiplies with the original value to leave a remainder of 1 under a chosen modulus.

How to Find a Modular Inverse

  1. Choose an integer and a modulus greater than 1.
  2. Run the extended Euclidean algorithm.
  3. Check that the greatest common divisor is 1.
  4. Normalize the coefficient to a positive representative if needed.
ax + my = 1

Worked Example

Example: 3 mod 11 has inverse 4 because 3 × 4 = 12 ≡ 1 mod 11.

3 × 4 mod 11 = 1

Frequently Asked Questions

When does a modular inverse exist?

It exists only when the number and modulus are coprime.

Can the inverse be negative?

The calculator normalizes the result into the standard positive representative.

Why is the modulus required to be greater than 1?

A modulus of 1 does not produce a meaningful modular inverse in this calculator.

Is this useful for RSA?

Yes. Modular inverses are a core part of RSA key generation and decryption.

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