# Natural-number divisibility

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

A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished.

mko-natural-divisibility · definition · v0.10.0



## Mathematical statement

For natural numbers a and b, a∣b exactly when there is a natural number k such that b=a·k. Every a divides 0, and 0∣b holds exactly when b=0.





## Representative formula (exact TeX)

b=m(a,k)





## Assumptions


- Natural numbers include 0 and multiplication is natural-number multiplication.
- The divisibility witness k must be natural; a noninteger quotient is not a witness.





## Symbol scope


- a — candidate divisor
- b — number being divided
- m — natural-number multiplication
- k — natural multiplier witness





## Intuition, definition, and misconceptions

When checking a∣b, first handle a=0: it holds only for b=0. For a>0 one may use quotient and remainder or search for k, but the remainder must be zero.

Every natural a divides 0 because 0=a·0. The relation 0∣0 also holds, and its multiplier witness is not unique.

Divisibility is reflexive and transitive. Finite exercises retain explicit multiplier witnesses rather than treating floating-point division or approximate equality as evidence.

A numerical value for b/a does not by itself prove a∣b, and a zero candidate divisor does not always make the divisibility relation false.





## Adopted derivation guide

[Natural-number divisibility: deciding by a multiplier witness, handling zero, and proving transitivity](https://ocme.evemisslab.com/en/derivations/derivation-mko-natural-divisibility-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Natural number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) — The domain and zero boundary come from the natural numbers.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — A divisibility witness is an exact multiplication equality.





## Verification boundary

Finite integer multiplication and quotient/remainder calculations can be recomputed exactly; checking instances does not formally prove every divisibility theorem.

not_formalized

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





## Next obligation

Bind the existential multiplier definition of Nat.Dvd, dvd_zero, zero_dvd and divisibility transitivity.





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