# Multiplicative Inverse Modulo Calculator

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

> The multiplicative inverse modulo calculator finds x with a·x ≡ 1 (mod m), or tells you that none exists, using the extended Euclidean algorithm.

## Inputs

- a
- m (modulus)

## Results

- Inverse x
- Check
- Extended Euclidean algorithm

## Formula

```
a·x ≡ 1 (mod m)  ⇔  a·x + m·y = 1 for some integer y
```

## Explanation

In modular arithmetic, dividing by a means multiplying by its inverse. Computing inverses is a daily task in cryptography – for example finding the private exponent d in RSA.

### How to use the Multiplicative Inverse Modulo Calculator

- **a**
- **m (modulus)**

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 the Bézout coefficients of a and m. If gcd(a, m) = 1, the coefficient of a, reduced modulo m, is the inverse.

## Worked example

Inputs — a: 17; m (modulus): 3,120.

Results — **Inverse x = 2753**; Check = 17 × 2753 = 46801 ≡ 1 (mod 3120); Extended Euclidean algorithm = Extended Euclidean algorithm:
17 = 0 × 3120 + 17
3120 = 183 × 17 + 9
17 = 1 × 9 + 8
9 = 1 × 8 + 1
8 = 8 × 1 + 0
Back-substitution gives 17 × (-367) + 3120 × (2) = 1, so x ≡ 2753 (mod 3120).

## Key terms

- **Bézout coefficients**: The integers x and y in ax + my = gcd(a, m).

## FAQ

**What is the inverse of 17 mod 3120?**
2753 – the classic RSA example.

**Why must a and m be coprime?**
Otherwise every multiple of a shares a factor with m and can never be 1 more than a multiple of m.

**Is the inverse unique?**
Yes, modulo m.

**What is Bézout's identity?**
For any a, m there are integers x, y with ax + my = gcd(a, m).


## Sources

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