# Integer mod inverse

`integer-mod-inverse` · version 1.0.0 · Numbers & math · free, no key needed

Compute the modular multiplicative inverse of a modulo modulus, when it exists.

**Use when you need to: integer mod inverse · modular multiplicative inverse · modular inverse.**

## Supported

- integer mod inverse
- modular multiplicative inverse
- modular inverse
- inverse mod m

## Not supported

- modular exponentiation
- extended gcd output
- inverse when gcd is not 1

## Behavior

- a and modulus are canonical integer strings; modulus must be a positive canonical integer.
- a may be negative or exceed modulus; it is reduced modulo modulus before inverting.
- The inverse exists only when gcd(a mod modulus, modulus) = 1; otherwise the request is rejected as undefined.
- modulus 1 is a degenerate ring where every residue is 0; this contract returns value 0 for modulus 1.
- The result is the unique value in [0, modulus) such that (a * value) mod modulus = 1 (or 0 when modulus is 1).
- Each input may have at most 1000 digits excluding sign.

## Input

- `a` (string, required): min length 1; max length 1001
- `modulus` (string, required): min length 1; max length 1001

## Output

- `value` (string, required)

## Limits

- max digits: 1000

## Example

Request input:

```json
{
  "a": "3",
  "modulus": "11"
}
```

Response:

```json
{
  "result": {
    "value": "4"
  }
}
```

## How to call it

### MCP

Connect `https://computefirst.net/mcp` ([setup](/docs#connect)), then call `execute` with:

```json
{
  "id": "integer-mod-inverse",
  "version": "1.0.0",
  "input": {
    "a": "3",
    "modulus": "11"
  }
}
```

### HTTP (no key)

```sh
curl -X POST https://computefirst.net/v1/tools/integer-mod-inverse/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"a":"3","modulus":"11"}'
```

The machine-readable contract is at [/v1/tools/integer-mod-inverse/versions/1.0.0](/v1/tools/integer-mod-inverse/versions/1.0.0).

### CLI

```sh
node cli.mjs run integer-mod-inverse 1.0.0 --input input.json --base-url https://computefirst.net
```

Get the client at [/clients/cli/](/clients/cli/).

## Related tools

- [Integer mod pow](/tools/integer-mod-pow): Compute base^exponent mod modulus exactly using fast modular exponentiation.
- [Integer CRT](/tools/integer-crt): Solve a system of integer congruences (x ≡ remainder mod modulus) with the Chinese Remainder Theorem.
- [Rational reciprocal](/tools/rational-reciprocal): Invert a rational number given as a numerator/denominator integer-string pair.
- [Array integer adjacent diff](/tools/array-integer-adjacent-diff): Replace an array of canonical integer strings with consecutive first differences.
- [Array integer clamp](/tools/array-integer-clamp): Clamp an array of canonical integer strings to optional inclusive min and max bounds.
- [Array integer cumulative sum](/tools/array-integer-cumulative-sum): Replace an array of canonical integer strings with equal-length prefix sums.