AMC12 2019 B

AMC12 2019 B · Q2

AMC12 2019 B · Q2. It mainly tests Logic puzzles, Primes & prime factorization.

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?

考虑命题：“如果 $n$ 不是质数，则 $n-2$ 是质数。”下列哪个 $n$ 的值是该命题的反例？

(A) 11 11

(B) 15 15

(D) 21 21

(E) 27 27

Answer

Correct choice: (E)

正确答案：（E）

Solution

An implication is false if and only if the hypothesis is true but the conclusion is false. Choice (E), $n = 27$, is a counterexample to the statement because the hypothesis is true (27 is not prime) but the conclusion is false (27 - 2 = 25 is not prime). For answer choices (A) and (C), $n$ is prime, so the hypothesis is false and these values of $n$ do not provide a counterexample. For choices (B), (C), and (D), $n - 2$ is prime, so the conclusion is true and these values of $n$ do not provide a counterexample.

蕴涵命题当且仅当前提为真而结论为假时为假。选项 (E)，$n = 27$ 是反例，因为前提为真（27 不是质数），但结论为假（27 - 2 = 25 不是质数）。对于选项 (A) 和 (C)，$n$ 是质数，故前提为假，这些值不是反例。对于选项 (B)、(C) 和 (D)，$n - 2$ 是质数，故结论为真，这些值不是反例。

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