Modular Inverse Calculator

What does this modular inverse calculator compute?

x is a modular inverse of a modulo m when a x ≡ 1 (mod m).

The published formula or method is ax ≡ 1 (mod m). The following references support the stated mathematics or method:

• MIT Mathematics for Computer Science — Chapter 8: Number Theory: Divisibility, prime factorization, gcd and the Euclidean algorithm, lcm via prime exponents, division with remainder, modular congruence and multiplicative inverses. CalculatorX’s signed-remainder and zero-input policies are stated separately on the page. Mathematical reference only; not validation or endorsement of this implementation.

Which inputs and limits should you check?

Use signed integer digit strings, at most48 digits and512 token characters; JSON numbers must be safe integers. Scientific notation, decimal points, commas, null, booleans and nested values are rejected. Unknown fields and conflicting aliases are errors. A nonzero modulus is required; inverse additionally needs |m|≥2 and gcd(a,m)=1.

The worked example uses 3·4 = 12 ≡ 1 with 3x ≡ 1 (mod 11). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.modulo v1.1.2 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Exact outputs and verification

All arithmetic uses BigInt, without floating rounding. Safe integer outputs are JSON numbers; larger exact outputs are signed decimal strings. The complete result includes the identity, algorithm, exactness model, work and calculation version. In inverse mode, the canonical Bézout coefficient x equals the returned inverse in [0,|m|), and y=(1−a·x)/m. These coefficients satisfy the original input identity a·x+m·y=1, including negative a and m.

For example, a=−3,m=11 gives inverse7, x=7,y=2: (−3)·7+11·2=1. Remainder/quotient obey a=q·m+r and0≤r<|m|. Integer changes can change these discrete outputs abruptly; floating conditioning and measurement uncertainty are not modeled. This implementation does not claim constant-time cryptographic behavior.

Inputs, sharing and remote calls

The main Modulo entry accepts remainder or inverse; the Modular Inverse discovery page and REST alias lock inverse, overriding a supplied mode. Share links use the current engine and do not preserve historical runtimes. Raw aliases are a/dividend and modulus/m/b; conflicting values reject rather than choosing one silently. Use digit strings for exact integers beyond the safe JSON numeric range. Errors include MISSING_REQUIRED_INPUT, INVALID_NUMBER, INVALID_MODE, UNKNOWN_INPUT, CONFLICTING_INPUT, VALUE_ABOVE_MAX, INPUT_TOO_LONG, DIVISION_BY_ZERO, INVALID_INPUT and NOT_INVERTIBLE.

Typing computes locally. Opening shared URLs and calling REST/MCP sends inputs to the server, where request URLs/metadata may enter logs. Recent calculations may remain in local browser history, which can be cleared. See the privacy policy; fixed retention, deletion guarantees and processing regions are not asserted here.