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Inverse Modulo Calculator

Select the type and enter the number in the designated field, and the calculator will instantly determine the inverse modulo, with detailed calculations shown.

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Inverse Modulo Calculator

Use this free inverse modulo calculator to quickly compute the modular inverse of any integer. It simplifies complex calculations for additive and multiplicative inverses.

What Is Inverse Modulo?

An integer x is called the inverse modulo of a given integer a if applying a specific operation between x and a produces the identity element.

Modulo Congruence

Two integers x and y are congruent modulo n if they have the same remainder when divided by n:

(x - y) is divisible by n → x ≡ y (mod n)

Generic representation: a ≡ b (mod n)

Types of Inverse Modulo

Additive Inverse Modulo

The additive identity is 0. The additive inverse modulo satisfies:

a + x ≡ 0 (mod m)

x is the additive inverse of a modulo m if a + x ≡ 0 (mod m)

Multiplicative Inverse Modulo

The multiplicative identity is 1. The modular multiplicative inverse satisfies:

a * x ≡ 1 (mod m)

x is the multiplicative inverse of a modulo m if a * x ≡ 1 (mod m)

Methods to Determine Multiplicative Inverse Modulo

1. Naive Method

Check all integers x ∈ {0, 1, ..., m-1}. The integer x that satisfies a * x ≡ 1 (mod m) is the multiplicative inverse. The calculator automates this efficiently.

2. Extended Euclidean Algorithm

Uses Bézout’s identity:

For integers a and m, if there exist x and y such that a * x + m * y = gcd(a, m),
and gcd(a, m) = 1 (a and m are coprime), then x is the multiplicative inverse of a modulo m.

Applying modulo m: a * x ≡ 1 (mod m)

3. Fermat’s Little Theorem

If m is prime and a is not divisible by m:

a^(m-1) ≡ 1 (mod m)

Thus, the multiplicative inverse of a modulo m is: a^(m-2) mod m

How the Inverse Modulo Calculator Works

Input:

Select either “Multiplicative Inverse” or “Additive Inverse”
Enter the integer and the modulus
Click “Calculate”

Output:

Calculates additive inverse modulo
Calculates multiplicative inverse modulo
Displays step-by-step calculations

FAQs

What is the inverse of 7 modulo 26?

The modular inverse is 15. You can verify this using the calculator.

What is an additive inverse?

The additive inverse is the negative of a number that, when added to the original number, yields 0 modulo m.

Generic representation: a + (-a) ≡ 0

What is the inverse of 13 modulo 2436?

The modular inverse is 927, which can also be verified with the calculator.

Conclusion

Inverse modulo is essential for solving linear congruences and cryptography problems. The inverse modulo calculator simplifies these calculations and provides accurate results with step-by-step guidance.

References

Wikipedia: Modular multiplicative inverse
Khan Academy: Modular inverses and exponentiation
Lumen Learning: Inverse Functions: Domain and Range

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