When does the modular inverse not exist?

The modular inverse of a (mod m) exists if and only if gcd(a, m) = 1. If a and m share a common factor greater than 1, no inverse exists and this tool will tell you so.

What is the extended Euclidean algorithm?

It extends the standard Euclidean algorithm (for computing GCDs) to also find integers x and y such that a·x + m·y = gcd(a, m). When gcd = 1, x is the modular inverse of a mod m.

Can I enter negative values for a?

Yes. The tool reduces a modulo m first, so negative values are handled correctly. For example, −3 mod 7 is treated as 4 mod 7.

Why is the result always in the range [0, m − 1]?

Modular inverses are defined up to multiples of m. This tool returns the canonical representative — the unique value in [0, m − 1] — so the answer is unambiguous.

How is this useful in cryptography?

RSA decryption requires computing the private exponent d = e⁻¹ mod φ(n). Similarly, elliptic-curve point division and many other cryptographic primitives rely on modular inverses.

Modular Inverse Calculator

Find the modular multiplicative inverse of a number mod m, with full extended Euclidean algorithm step-by-step.

Inputs

Integer a

Any integer (positive or negative).

Modulus m

Must be an integer greater than 1.

Quick Examples

Enter values and click Calculate.

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Summary

Find the modular multiplicative inverse of a number mod m, with full extended Euclidean algorithm step-by-step.

How it works

Enter the integer a whose inverse you want to find.
Enter the modulus m (must be greater than 1).
Click Calculate to run the extended Euclidean algorithm.
The tool checks that gcd(a, m) = 1 — a necessary condition for the inverse to exist.
Each step of the back-substitution is shown so you can follow the full working.
The final result x is the unique inverse in the range [0, m − 1].

Use cases

Solving linear congruences in number theory coursework.
Computing private keys and inverses in RSA cryptography.
Working through Diffie-Hellman key exchange problems.
Checking homework answers for modular arithmetic assignments.
Computing modular division (a / b mod m = a · b⁻¹ mod m).
Understanding the extended Euclidean algorithm step by step.
Preparing for competitive programming contests involving number theory.
Verifying implementations of modular arithmetic in code.

Frequently Asked Questions

Last updated: 2026-09-22 · Reviewed by Nham Vu