# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Natural-number divisibility: deciding by a multiplier witness, handling zero, and proving transitivity](/en/derivations/derivation-mko-natural-divisibility-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-natural-divisibility)

From the MKO's definition (a∣b ⇔ there is a natural k with b=a·k): (1) prove 6∣84; (2) prove 6∤87; (3) prove 4∤10, although 10/4=5/2 is a number; (4) prove that every a divides 0; (5) prove 0∣b ⇔ b=0, and explain why the witness for 0∣0 is not unique and why 0∤5; (6) prove that for a>0 there is at most one witness (and when a∣b that existing witness is unique); (7) prove reflexivity and transitivity of divisibility, and from 3∣6 and 6∣84 find a witness for 3∣84.

Natural-number divisibility

derivation-mko-natural-divisibility-01





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