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2019 AMC 10B Problem 2

2019 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Logic Puzzles · difficulty 1 of 5 · about 1 min · position P1-10

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?

A)
$11$
B)
$15$
C)
$19$
D)
$21$
E)
$27$

Answer

Key insight

A counterexample must make the hypothesis true and the conclusion false: n composite and n-2 composite, which only 27 (with 25) does.

Solution

A counterexample to "if $P$ then $Q$ " is a case where $P$ holds but $Q$ fails. Here we need $n$ to be not prime while $n - 2$ is also not prime.

Check the choices:

$11$ and $19$ are prime, so the hypothesis is false and they cannot be counterexamples.
$15$ is composite, but $15 - 2 = 13$ is prime: the statement holds.
$21$ is composite, but $21 - 2 = 19$ is prime: the statement holds.
$27$ is composite, and $27 - 2 = 25 = 5^2$ is composite: the statement fails.

The answer is $\boxed{\textbf{(E)}\ 27}$ .

Why this works

An implication is only broken when its "if" part is satisfied and its "then" part is not. Testing each choice against both conditions is quick, and the prime choices can be discarded immediately because they never trigger the hypothesis. The same two-part check applies to every "which is a counterexample" question.

The trap

Choosing a prime n such as 11 or 19; when n is prime the hypothesis is false, so the statement is not violated.

Common mistakes

Choosing a prime n such as 11 or 19; when n is prime the hypothesis is false, so the statement is not violated.
Checking only whether $n - 2$ is prime and forgetting to require that $n$ itself be composite.

Techniques

Use the answer choices (mod checks, size, form) to eliminate or select · Test small/specific values or special cases to find or verify the answer