# Integer digital root

`integer-digital-root` · version 1.0.0 · Numbers & math · free, no key needed

Repeatedly sum the base-b digits of the magnitude of an integer string until a single digit remains.

**Use when you need to: integer digital root · repeated digit sum · digital root in a base.**

## Supported

- integer digital root
- repeated digit sum
- digital root in a base

## Not supported

- single-pass digit sum
- digital root of a fraction
- casting out nines shortcut only

## Behavior

- value is a canonical integer string: 0 or an optional ASCII minus followed by a non-zero leading digit, at most 1000 digits excluding sign.
- base is a JSON integer from 2 to 36; when omitted it defaults to 10.
- The sign of value is ignored: the digital root is computed on the magnitude |value|.
- value 0 has digital root 0 in every base.
- The base-b digit sum is repeated until the result is a single base-b digit, i.e. strictly less than base.
- digital_root is returned as a JSON integer in [0, base).

## Input

- `value` (string, required): max length 1001
- `base` (integer, optional): min 2; max 36; default 10

## Output

- `digital_root` (integer, required): min 0

## Limits

- max digits: 1000
- min base: 2
- max base: 36

## Example

Request input:

```json
{
  "value": "9875"
}
```

Response:

```json
{
  "result": {
    "digital_root": 2
  }
}
```

## How to call it

### MCP

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

```json
{
  "id": "integer-digital-root",
  "version": "1.0.0",
  "input": {
    "value": "9875"
  }
}
```

### HTTP (no key)

```sh
curl -X POST https://computefirst.net/v1/tools/integer-digital-root/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"value":"9875"}'
```

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

### CLI

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

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

## Related tools

- [Integer digit sum](/tools/integer-digit-sum): Sum the base-b digits of the magnitude of an integer string.
- [Array integer cumulative sum](/tools/array-integer-cumulative-sum): Replace an array of canonical integer strings with equal-length prefix sums.
- [Array integer GCD](/tools/array-integer-gcd): Compute the non-negative greatest common divisor of an array of canonical integer strings.
- [Array integer LCM](/tools/array-integer-lcm): Compute the non-negative least common multiple of an array of canonical integer strings.
- [Array integer sum](/tools/array-integer-sum): Sum an array of canonical integer strings and return the canonical total.
- [Array rational sum](/tools/array-rational-sum): Sum an array of canonical rationals and return the reduced total.