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Multiplicative Inverse Modulo Calculator.

Find the modular inverse of an integer when it exists.

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Set your values

Results update as you type.

Result: inverse: 4.00000000 · modulus: 11.00000000

Result

inverse: 4.00000000 · modulus: 11.00000000

Use Cases

Solve modular equations

Use the inverse to solve linear congruences like ax ≡ b (mod m) by multiplying both sides by the inverse of a.

Example: Solve 3x ≡ 4 (mod 7) by finding inverse of 3 mod 7 (which is 5) to get x ≡ 20 ≡ 6 (mod 7).

Cryptography and number theory

Modular inverses are essential in RSA encryption, hashing, and other cryptographic algorithms that rely on modular arithmetic.

Example: Compute the private key exponent in RSA by finding the modular inverse of e modulo φ(n).

Frequently Asked Questions

What is a multiplicative inverse modulo m?
A multiplicative inverse of a modulo m is an integer x such that (a * x) mod m = 1. It exists only if a and m are coprime (gcd(a, m) = 1).
How do I use this calculator?
Enter the integer value (a) and the modulus (m). The calculator will compute the inverse if it exists, or indicate that it does not exist when gcd(a, m) ≠ 1.
What if the inverse does not exist?
If a and m are not coprime, the modular inverse does not exist. The calculator will show that no inverse exists, so you can adjust your inputs.

Tips & Common Mistakes

Tips

  • Ensure the modulus m is a positive integer greater than 1.
  • Check if a and m are coprime: the inverse exists only if their greatest common divisor is 1.
  • The inverse is unique modulo m; the calculator returns the smallest positive representative.
  • Use the result to verify: multiply a by the inverse, and the product should be congruent to 1 modulo m.

Common Mistakes to Avoid

  • Assuming the inverse always exists; it only exists when gcd(a, m) = 1.
  • Entering a negative value for the modulus; m must be positive.
  • Confusing the modular inverse with the reciprocal in real numbers; the inverse is an integer modulo m.

Last updated: August 13, 2026