14. Which of the following values of the positive integer $n$ is a counterexample to the statement: "If $n$ is not prime then $n-2$ is not prime"?
A 6
B 11
С 27
D 33
E 51
SOLUTION
D
A counterexample to the statement "If $n$ is not prime then $n-2$ is not prime" is a positive integer $n$ for which this implication is false. An implication of the form "If $P$ then $Q$ " is false if $P$ is true, but $Q$ is false.

Therefore a counterexample to the statement is a positive integer $n$ for which it is true that $n$ is not prime, but false that $n-2$ is not prime. In other words, a counterexample is a positive integer $n$ such that $n$ is not prime but $n-2$ is prime.

Since 11 is prime, option B does not provide the required counterexample.
The options $6,27,33$ and 51 are not primes. However $6-2=4,27-2=25$ and $51-2=49$ are also not primes. Therefore none of the options $\mathrm{A}, \mathrm{C}$ and $\mathrm{E}$ provides the required counterexample.
This leaves $n=33$. We see that 33 is not prime, but $33-2=31$, is prime. Therefore 33 is a counterexample. Hence the correct option is D.