AMC 10 · 2019 · #2
Grade 4 logicPick an answer.
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.
Cross out the primes
Prime choices are out at once.
If n is prime, the 'if' part of the claim is false and there's nothing to break.
If a number is prime, the claim's condition is not met and there is nothing to break.
▸ Why?
A counterexample must meet the condition and miss the promise, so failing the condition disqualifies it.
▸ Why?
A claim only speaks about the cases inside its condition, so cases outside it say nothing at all.
Subtract two from the survivors
Subtract two from the three left.
Subtract 2 from each survivor and check the result.
4.NBT.B.4Make A Systematic ListPick the non-prime result
Only one result is composite, so the answer is 27.
Only one survivor leaves n - 2 composite — and that one breaks the claim.
4.OA.B.4Eliminate PossibilitiesThis AMC 12 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).