# Concept derivation guides

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

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Concept guide



## [Prime natural numbers: proving 97 prime and 91 composite, and excluding 0 and 1](/en/derivations/derivation-mko-prime-number-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-prime-number)

From the MKO's definition (p is prime ⇔ 2≤p and every natural divisor d of p satisfies d=1 or d=p): (1) explain why 0 and 1 are not prime; (2) prove that 2 is prime; (3) prove that 91 is composite with the witness 7, and show that testing only 2, 3, 5 misjudges it; (4) prove 'if n=dk with d, k≥2 then min(d,k)²≤n', and use it to prove that 97 is prime by checking only 2 to 9; (5) prove that 2 is the only even prime.

Prime natural number

derivation-mko-prime-number-01





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