2019 AMC 10B Problem 2

P1-10~1 minPrint

Logic Puzzles Primes 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

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?

Show solution without answering

Key insight

To break an if-then statement, the if part must be true and the then part false. So n and n minus 2 must both be not prime.

Solution

The statement says: "If $n$ is not prime, then $n - 2$ is prime." A counterexample is a number that makes the "if" part true and the "then" part false. So we need both of these:

$n$ is not prime, and
$n - 2$ is also not prime.

Check each choice.

$n$	Is $n$ prime?	$n - 2$	Is $n - 2$ prime?	Breaks the statement?
$11$	yes	$9$	no	no ("if" part is false)
$15$	no ( $3 \cdot 5$ )	$13$	yes	no
$19$	yes	$17$	yes	no ("if" part is false)
$21$	no ( $3 \cdot 7$ )	$19$	yes	no
$27$	no ( $3 \cdot 9$ )	$25$	no ( $5 \cdot 5$ )	yes

Only $27$ makes the "if" part true while the "then" part fails.

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

Why this works

An if-then statement only makes a promise when its "if" part happens. A prime $n$ never triggers that promise, so it can never be a counterexample, even though $11 - 2 = 9$ is not prime. For any "which is a counterexample" question, write down the two conditions separately and test each choice against both.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(E)}\ 27}$ .

Review in the AoPS books

Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.4)
Introduction to Number Theory, Chapter 2: Primes and Composites
Introduction to Number Theory, Chapter 4: Prime Factorization (§4.2, §4.4)