10. Which of the following five values of $n$ is a counterexample to the statement in the box below?
For a positive integer $n$, at least one of $6 n-1$ and $6 n+1$ is prime.
A 10
B 19
C 20
D 21
E 30
Solution
C
A counterexample to the statement in the box is a value of $n$ for which it is not true that at least one of $6 n-1$ and $6 n+1$ is prime. That is, it is a value of $n$ for which neither $6 n-1$ nor $6 n+1$ is prime.
We set out the values of $6 n-1$ and $6 n+1$ for $n=10,19,20,21$ and 30 in the following table.
The values of $6 n-1$ and $6 n+1$ that are not prime are shown in bold.
We therefore see that for $n=20$, neither $6 n-1$ nor $6 n+1$ is prime. Therefore $n=20$ provides the required counterexample.