Inverse Modulo Calculator

Name: Inverse Modulo Calculator
Author: Jitendra Kumar

Calculate modular inverses with the extended Euclidean algorithm, Bezout identity, gcd checks, and target congruence solving.

Last Updated: May 2026

Modular inverse

GCD

Normalized value

Check

1 mod 7

Inverse Modulo Inputs

Enter an integer value and a modulus greater than 1. The calculator normalizes the value, runs extended Euclid, and verifies whether the inverse exists.

Value (a)

Modulus (m)

Target (b)

Optional context for solving a x = b mod m.

Inverse Proof

Check	Calculation	Result
Congruence	3 x inverse = 1 mod 7	Solvable
Coprime test	gcd(3, 7)	1
Bezout identity	3 x -2 + 7 x 1	1
Inverse check	3 x 5 mod 7	1

Target Congruence

Item	Calculation	Result
Target congruence	3 x = 1 mod 7	Solvable
Solution	x = inverse x target mod 7	5
Verification	3 x 5 mod 7	1

Euclidean Algorithm Steps

Division step	Note
3 = 7 x 0 + 3	Continue with divisor and remainder.
7 = 3 x 2 + 1	Continue with divisor and remainder.
3 = 1 x 3 + 0	GCD is 1

Number Theory Notice

This calculator is for integer modular arithmetic and educational number theory. For cryptographic production systems, use audited libraries and verified parameter handling rather than manual calculations.

Checked by Jitendra Kumar

Inverse Modulo Calculator is checked for formula labels, source links, and result limits.

Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.

Sources & methodology · Review standards

How to Use the Inverse Modulo Calculator

Enter the value a and modulus m. The modulus must be greater than 1. The calculator reduces a modulo m, checks gcd(a,m), and reports the inverse when it exists.

Use the target field when you want to solve a linear congruence in the form a x = b mod m. When the inverse exists, the solution is inverse times b modulo m.

Step 1: Enter the value
This is the integer whose modular inverse you want.
Step 2: Enter the modulus
Use an integer modulus greater than 1.
Step 3: Check the inverse result
The calculator shows the inverse only when gcd(value, modulus) is 1.
Step 4: Review the proof
Use the Bezout identity and Euclidean steps to verify why the inverse exists or fails.

How This Inverse Modulo Calculator Works

A modular inverse is the number that turns multiplication into 1 within a modulus. If x is the inverse of a modulo m, then a x has remainder 1 after division by m.

The inverse exists only when a and m are coprime. The calculator proves that condition with the extended Euclidean algorithm, which finds coefficients in the equation a x + m y = gcd(a,m).

When the gcd equals 1, the coefficient of a is normalized into the range 0 through m - 1 and returned as the modular inverse.

Modular Inverse Guide

Core Formulas

Concept	Formula	Use
Modular inverse	a x a^-1 = 1 mod m	Defines the inverse of a modulo m.
Existence test	gcd(a, m) = 1	An inverse exists exactly when a and m are coprime.
Bezout identity	a x + m y = gcd(a, m)	Extended Euclid finds the coefficient x.
Inverse from Bezout	a x + m y = 1 implies a x = 1 mod m	Normalize x into 0 through m - 1.
Modular division	a x = b mod m, so x = a^-1 b mod m	Works when the inverse exists.

Examples

Problem	Result	Notes
3 inverse mod 7	5	3 x 5 = 15 = 1 mod 7.
17 inverse mod 3120	2,753	Common RSA-style private exponent example.
12 inverse mod 18	Does not exist	gcd(12,18) = 6, so the inverse test fails.
-11 inverse mod 26	7	-11 is normalized to 15, and 15 x 7 = 1 mod 26.

Modular Inverse Context

Modular inverses let you divide inside modular arithmetic. Since direct division is not generally available modulo m, multiplying by an inverse fills that role when the inverse exists.

This is why the coprime check matters. Prime moduli make every nonzero residue invertible, while composite moduli have values that fail the gcd test.

Keep the research moving with Modulo Calculator, LCM Calculator, LCM / GCF Calculator, and Long Multiplication Calculator.

Frequently Asked Questions

A modular inverse of a modulo m is an integer x such that a times x leaves a remainder of 1 when divided by m.

A modular inverse exists exactly when the value and modulus are coprime, meaning their greatest common divisor is 1.

It uses the extended Euclidean algorithm to find integers x and y where a x + m y equals gcd(a,m). When the gcd is 1, x is the inverse modulo m after normalization.

Yes. The calculator first reduces the value into the standard modulo range from 0 to m - 1, then computes the inverse.

If a and m share a factor larger than 1, every product a x shares that factor modulo m, so it cannot be congruent to 1.

Modular inverses are used in modular division, solving linear congruences, cryptography, RSA examples, hashing, and number theory.