# Modular inverse

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

For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n.

mko-modular-inverse · definition · v0.10.0



## Mathematical statement

Use extended Euclid for an exact Bezout certificate a*x+n*y=1 and return the normalized representative 0≤x mod n<n. Modulus zero is invalid, and this slice requires n≥2. A nonunit explicitly has no inverse; approximate division is not used.





## Representative formula (exact TeX)

I(a)





## Assumptions


- a is an integer and n≥2 is the modulus.
- Congruence and gcd calculations use exact integer arithmetic.





## Symbol scope


- I — modular inverse
- a — integer representative





## Intuition, definition, and misconceptions

gcd(a,n)=1 is equivalent to inverse existence.

Extended Euclid supplies a Bezout witness.

Return a normalized residue; a nonunit has no inverse.

Do not divide by a nonunit or use modulus zero, and do not confuse a unique residue class with a unique integer representative.





## Adopted derivation guide

[The normalized inverse of 17 modulo 43](https://ocme.evemisslab.com/en/derivations/derivation-mko-modular-inverse-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Greatest common divisor and coprimality](https://ocme.evemisslab.com/en/mko/mko-gcd-coprime/) — The gcd condition and Bezout identity characterize existence.
- [Natural-number congruence modulo a positive modulus](https://ocme.evemisslab.com/en/mko/mko-modular-congruence/) — Uniqueness is only up to a residue class.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) — The certificate satisfies a*x+n*y=1.





## Verification boundary

This slice requires n≥2 and does not adopt a degenerate n=1 convention.

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 Chinese remainder theorem and modular linear equations.





## 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/modular-inverse.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-modular-inverse/index.md)