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

Name: Modular Inverse Calculator
Availability: InStock
Author: Nham Vu

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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