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Modular multiplicative inverse calculator

Find a⁻¹ mod m — the number x with a·x ≡ 1 (mod m) — using the extended Euclidean algorithm. An inverse exists exactly when gcd(a, m) = 1; otherwise the tool reports that none exists. The modular inverse is what lets you 'divide' in modular arithmetic and is the step that produces the RSA private exponent d from the public exponent e.

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Enter a, Modulus m and the Modular multiplicative inverse calculator works out a⁻¹ mod m, gcd(a, m), Verification straight away. For instance, with a = 3 and Modulus m = 7 it returns a⁻¹ mod m = 5, gcd(a, m) = 1 and Verification = 3 × 5 = 15 ≡ 1 (mod 7).

How to use it

  1. Enter your values: a, Modulus m.
  2. Read the result instantly: a⁻¹ mod m, gcd(a, m), Verification.

Frequently asked questions

What does the Modular multiplicative inverse calculator actually compute?

It takes a and Modulus m and derives a⁻¹ mod m, gcd(a, m) and Verification from them. The calculation is live as you type, so the result updates on every change.

What information do I need to provide?

2 values: a and Modulus m. Nothing else is required — no account, no file upload.

Can you show a worked example?

With a = 3 and Modulus m = 7, the calculator returns a⁻¹ mod m = 5, gcd(a, m) = 1 and Verification = 3 × 5 = 15 ≡ 1 (mod 7). Those figures come from running this exact tool, so you can reproduce them by entering the same values.

What happens if I enter larger values?

It moves a lot. Using a = 6 and Modulus m = 14 instead, gcd(a, m) goes from 1 to 2 — which is why it is worth testing a few scenarios rather than trusting a single figure.

When would I actually use this?

Reading a number written another way: a Roman numeral on a monument, a hexadecimal colour, a binary byte, or a fraction that needs reducing.

What is the most common mistake?

Assuming a decimal fraction survives a change of base. A tenth is exact in decimal and infinitely repeating in binary, which is why 0.1 + 0.2 is not 0.3 in most programming languages.

What is the difference between the Modular multiplicative inverse calculator and the Modular exponentiation calculator?

This one returns a⁻¹ mod m and gcd(a, m); the Modular exponentiation calculator returns aᵇ mod n and Base reduced (a mod n). That is the whole difference — open the one whose figure you need.

Is there a tool for the next step?

Antilog Calculator (inverse logarithm) is the closest one after this: The antilogarithm undoes the logarithm: antilog_b(y) = bʸ, so log_b(bʸ) = y. Pick base 10, e, 2 or your own, enter the exponent, and the result comes back with the inverse relationship spelled out and the round trip checked. Bases of zero or less, and base 1, are rejected with the reason.

What else is worth having open alongside it?

Inverse Function Calculator and Inverse Laplace Transform Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from, and how current are they?

Base conversion and number-theory results are exact. Roman numerals follow the standard subtractive form used since the Middle Ages, which is not the only one the Romans themselves used.

Further reading

All guides →
How-toLCM and GCD ExplainedThe GCD is the biggest shared divisor; the LCM is the smallest shared multiple. Here's what each means, how to find them with prime factors, and the shortcut linking them.How-toHow to Simplify a RatioDivide both sides by their greatest common divisor. Here's what a ratio is, how to reduce it, why units must match first, and how to compare ratios.How-toHow to Round Numbers: Round Half Up, Round Half to Even, and MoreRound to any decimal place or number of significant figures, and learn when to use round half up versus round half to even to avoid bias.ExplainerSignificant Figures Explained: Counting, Rounding, and Why They MatterLearn what counts as a significant figure, the rules for leading, trailing, and captive zeros, and how sig figs keep your calculated answers honest.How-toHow to Find Prime Factors: Factor Trees and the Division MethodBreak any whole number into its prime building blocks with a factor tree or repeated division — and write the result as a clean prime factorization with exponents.How-toHow to Multiply and Divide Fractions: Multiply Across, Flip to DivideMultiplying fractions needs no common denominator — just multiply across. Dividing means multiplying by the reciprocal. Learn both with worked examples and how to simplify.