11. Alitta claims that if $p$ is an odd prime then $p^{2}-2$ is also an odd prime. Which of the following values of $p$ is a counterexample to this claim?
A 3
B 5
C 7
D 9
E 11
Solution
A counterexample to the claim is an odd prime $p$ such that $p^{2}-2$ is not an odd prime. 3 is an odd prime, and $3^{2}-2=7$ is also an odd prime. So 3 is not a counterexample. 5 is an odd prime, and $5^{2}-2=23$ is also an odd prime. So 5 is not a counterexample. 7 is an odd prime, and $7^{2}-2=47$ is also an odd prime. So 7 is not a counterexample. 9 is not an odd prime. So 9 is not a counterexample 11 is an odd prime, but $11^{2}-2=119=7 \times 17$ is not an odd prime. Therefore 11 is a counterexample.