Integer CRT

integer-crt · version 1.0.0 · Numbers & math · free, no key needed

Solve a system of integer congruences (x ≡ remainder mod modulus) with the Chinese Remainder Theorem.

Use when you need to: integer crt · chinese remainder theorem · solve system of congruences.

Supported

• integer crt
• chinese remainder theorem
• solve system of congruences
• combine modular congruences

Not supported

• polynomial congruences
• non-integer moduli
• inconsistent system resolution

Behavior

• congruences is an array of 1 to 100 objects, each with canonical integer-string remainder and a positive canonical integer-string modulus.
• remainder may be negative or exceed modulus; it is reduced into [0, modulus) before solving.
• Moduli need not be pairwise coprime; congruences are merged pairwise with a consistency check at each step.
• If any pair of congruences is inconsistent (no integer satisfies both), the request is rejected as unsolvable.
• The output is the unique remainder in [0, combined_modulus) together with combined_modulus, the least common multiple of every input modulus.
• A single congruence returns its own reduced remainder and modulus unchanged.
• Every modulus and the combined modulus may have at most 1000 digits excluding sign.

Input

• congruences (array of object, required): min items 1; max items 100

Output

• remainder (string, required)
• modulus (string, required)

Limits

• max congruences: 100
• max digits: 1000

Example

Request input:

{
  "congruences": [
    {
      "remainder": "2",
      "modulus": "3"
    },
    {
      "remainder": "3",
      "modulus": "5"
    }
  ]
}

Response:

{
  "result": {
    "remainder": "8",
    "modulus": "15"
  }
}

How to call it

MCP

Connect https://computefirst.net/mcp (setup), then call execute with:

{
  "id": "integer-crt",
  "version": "1.0.0",
  "input": {
    "congruences": [
      {
        "remainder": "2",
        "modulus": "3"
      },
      {
        "remainder": "3",
        "modulus": "5"
      }
    ]
  }
}

HTTP (no key)

curl -X POST https://computefirst.net/v1/tools/integer-crt/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"congruences":[{"remainder":"2","modulus":"3"},{"remainder":"3","modulus":"5"}]}'

The machine-readable contract is at /v1/tools/integer-crt/versions/1.0.0.

CLI

node cli.mjs run integer-crt 1.0.0 --input input.json --base-url https://computefirst.net

Get the client at /clients/cli/.

Related tools

• Array integer adjacent diff: Replace an array of canonical integer strings with consecutive first differences.
• Array integer cumulative sum: Replace an array of canonical integer strings with equal-length prefix sums.
• Array integer GCD: Compute the non-negative greatest common divisor of an array of canonical integer strings.
• Array integer LCM: Compute the non-negative least common multiple of an array of canonical integer strings.
• Array integer median: Select lower and upper middle canonical integers; emit median only for odd length.
• Array integer min max: Return the numeric minimum and maximum of canonical integer strings.