Natural-number divisibility

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Natural number — The domain and zero boundary come from the natural numbers.
• 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.