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Modular inverse calculator

Solve a × x ≡ 1 (mod m) for the least non-negative integer x, or see why no inverse exists.

Enter two integers

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Negative and zero values are allowed; up to 200 digits.

Use a positive integer greater than 1; up to 200 digits.

Modular inverse result

Enter a and a modulus greater than 1, then calculate.

How to find a modular inverse

A modular inverse undoes multiplication within one modulus. This page handles the multiplicative inverse only; it does not generate RSA keys or factor numbers.

How to use

  1. Enter any integer a, including a negative value.
  2. Enter a positive modulus m greater than 1.
  3. Calculate to get the unique representative x from 0 through m − 1, when it exists.
  4. Read the Bézout line and the actual product remainder to verify the answer independently.

Condition and formula

An inverse exists exactly when gcd(a, m) = 1. The extended Euclidean algorithm finds coefficients u and v with a·u + m·v = 1; u reduced modulo m is the inverse.

Negative a is first normalized as ((a mod m) + m) mod m. The displayed answer is always the least non-negative representative.

Worked examples

3 modulo 11

gcd(3, 11) = 1, and 3 × 4 = 12. Because 12 mod 11 = 1, the inverse is 4.

−3 modulo 11

−3 normalizes to 8. Since 8 × 7 = 56 and 56 mod 11 = 1, the inverse is 7.

6 modulo 9

gcd(6, 9) = 3, so no integer multiplied by 6 can leave remainder 1 modulo 9.

Input limits and scope

  • Each input is limited to 200 decimal digits to keep browser work responsive. Leading zeros still count toward the typed limit.
  • Only base-10 integers are accepted. Decimals, scientific notation, spaces inside a number, typed units, and Infinity are rejected.
  • A modulus of 0, 1, or a negative number is invalid. a = 0 is accepted but has no inverse for m > 1.
  • The result is an exact integer, not a decimal approximation. This calculator is an arithmetic aid, not a cryptographic key generator or security service.

Frequently asked questions

Why must the GCD equal 1?

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

Can a negative integer have an inverse?

Yes. It is reduced to its least non-negative residue before the same coprimality test.

Why is there only one displayed answer?

All inverses differ by a multiple of m. The calculator displays the unique one in the range 0 to m − 1.

Is this the same as 1/a?

No. A modular inverse is an integer whose product has remainder 1 under a chosen modulus.