# Inverse Modulo Calculator

URL: https://speedycalculator.com/calculators/math/inverse-modulo/
Last updated: 2026-10-30

> The inverse modulo calculator finds the modular multiplicative inverse (a·x ≡ 1 mod m) with the extended Euclidean algorithm, or the modular additive inverse (a + x ≡ 0 mod m).

## Inputs

- Inverse type
- a
- Modulus m

## Results

- Inverse x
- Check
- Steps

## Formula

```
Multiplicative: a·x ≡ 1 (mod m), exists ⇔ gcd(a, m) = 1
Additive: x ≡ −a (mod m)
```

## Explanation

Modular inverses are the "division" of modular arithmetic. They are essential in RSA and elliptic-curve cryptography, in the Chinese remainder theorem and in hash functions and random number generators.

### How to use the Inverse Modulo Calculator

- **Inverse type**
- **a**
- **Modulus m**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The extended Euclidean algorithm finds integers x and y with a·x + m·y = gcd(a, m). When the gcd is 1, x (reduced modulo m) is the inverse. The table of divisions is shown so you can follow the steps.

## Worked example

Inputs — Inverse type: Modular multiplicative inverse; a: 3; Modulus m: 11.

Results — **Inverse x = 4**; Check = 3 × 4 = 12 ≡ 1 (mod 11); Steps = Extended Euclidean algorithm:
3 = 0 × 11 + 3
11 = 3 × 3 + 2
3 = 1 × 2 + 1
2 = 2 × 1 + 0
Back-substitution gives 3 × (4) + 11 × (-1) = 1, so x ≡ 4 (mod 11).

## Key terms

- **Modular inverse**: A number that multiplies with a to give 1 modulo m.

## FAQ

**What is the inverse of 3 mod 11?**
4, because 3 × 4 = 12 ≡ 1 (mod 11).

**When does a multiplicative inverse not exist?**
When a and m share a factor, e.g. 2 has no inverse mod 4.

**What is the additive inverse of 3 mod 11?**
8, because 3 + 8 = 11 ≡ 0.

**Can I use Fermat's little theorem?**
If m is prime, a^(m−2) mod m is the inverse.


## Sources

- [Wolfram MathWorld – Modular inverse](https://mathworld.wolfram.com/ModularInverse.html)
- [OpenStax – Contemporary Mathematics](https://openstax.org/details/books/contemporary-mathematics)