# Chinese remainder theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product.

mko-chinese-remainder-theorem · theorem · v0.10.0



## Mathematical statement

Normalize residues, construct a simultaneous solution using modular-inverse methods and check every congruence. Uniqueness is modulo the product, not uniqueness as an integer. Modulus zero is invalid. A noncoprime system is outside the main hypothesis but may be compatible; test pairwise gcd conditions and use the lcm period in general.





## Representative formula (exact TeX)

C(R)





## Assumptions


- The main CRT questions use positive pairwise-coprime moduli.
- All integer arithmetic is exact.





## Symbol scope


- C — CRT solution
- R — residue system





## Intuition, definition, and misconceptions

Construct a normalized simultaneous solution.

Check every congruence.

Noncoprime systems need compatibility checks and are not automatically inconsistent.

Unique modulo the product does not mean a unique integer, and noncoprimality alone does not imply inconsistency.





## Adopted derivation guide

[Constructing a three-modulus CRT class](https://ocme.evemisslab.com/en/derivations/derivation-mko-chinese-remainder-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Integer](https://ocme.evemisslab.com/en/mko/mko-integer/) — Solutions and residues are integers.
- [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/) — Each system equation is a congruence.
- [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/mko/mko-gcd-coprime/) — The main hypothesis is pairwise coprimality.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Every equation and the residue-class uniqueness obligation must hold.





## Verification boundary

Keep the pairwise-coprime theorem separate from noncoprime boundary audits.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add the generalized Chinese remainder theorem.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/chinese-remainder-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-chinese-remainder-theorem/index.md)