{"id":"integer-crt","version":"1.0.0","description":"Solve a system of integer congruences (x ≡ remainder mod modulus) with the Chinese Remainder Theorem.","supported_operations":["integer crt","chinese remainder theorem","solve system of congruences","combine modular congruences"],"unsupported_operations":["polynomial congruences","non-integer moduli","inconsistent system resolution"],"semantics":["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."],"limits":{"max_congruences":100,"max_digits":1000},"pricing":{"status":"unpriced","charge_usd":null},"input_schema":{"type":"object","additionalProperties":false,"required":["congruences"],"properties":{"congruences":{"type":"array","minItems":1,"maxItems":100,"items":{"type":"object","additionalProperties":false,"required":["remainder","modulus"],"properties":{"remainder":{"type":"string","maxLength":1001},"modulus":{"type":"string","maxLength":1001}}}}}},"output_schema":{"type":"object","additionalProperties":false,"required":["remainder","modulus"],"properties":{"remainder":{"type":"string"},"modulus":{"type":"string"}}},"examples":[{"input":{"congruences":[{"remainder":"2","modulus":"3"},{"remainder":"3","modulus":"5"}]},"output":{"remainder":"8","modulus":"15"}}],"execute_url":"/v1/tools/integer-crt/versions/1.0.0/execute"}