AMC 10 · 2019 · #2

Grade 4 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

prime-numbersprimality-testlogical-deduction caseworksystematic-enumeration ↑ Prerequisites: prime-numbersprimality-test

📏 Short solution 💡 2 insights

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Problem

Consider the claim: 'If n is not prime, then n - 2 is prime.' Among the five answer choices, find a value of n that breaks this claim — that is, n is not prime BUT n - 2 is also not prime.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Tool #3 (Eliminate): the answer must satisfy two conditions. First, n itself must be composite, which kills any prime choice — that removes 11 and 19 immediately. Tool #2 (Systematic List): for each surviving composite n, compute n - 2 and check primality. The single survivor is the counterexample.

1STEP 1

First filter — keep only non-prime n: 11 and 19 are prime, so eliminate them; 15, 21, 27 are composite and survive.

11, 19 prime → eliminated. Survivors: 15, 21, 27.

2STEP 2

Second filter — subtract 2 from each survivor: 15 - 2 = 13, 21 - 2 = 19, 27 - 2 = 25; now test each result for primality.

15 - 2 = 13, 21 - 2 = 19, 27 - 2 = 25

3STEP 3

Test primality: 13 and 19 are prime (claim holds), but 25 = 5 · 5 is composite — so n = 27 breaks the claim.

13 prime, 19 prime, 25 = 5 · 5 composite → n = 27 → (E)

Answer

Verify the counterexample at n = 27. Is 27 not prime? Yes — 27 = 3 · 9 = 3³, so composite. Is 27 - 2 = 25 prime? No — 25 = 5 · 5, also composite. So the implication 'not prime → n - 2 prime' has a true premise but a false conclusion — exactly what a counterexample requires. The other composites 15 and 21 both give a prime n - 2, so they're consistent with the claim.

💡Key takeaway

This AMC 10 problem only needs Grade 4 prime-vs-composite checks you already know — toss out the primes (11, 19), then subtract 2 from each leftover and look for the first composite result. 27 - 2 = 25 = 5 · 5, so (E).

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