4. For how many positive integer values of $n$ is $n^{2}+2 n$ prime?
A 0
B 1
C 2
D 3
E more than 3
Solution
$\mathbf{B}$

We have $n^{2}+2 n=n(n+2)$. Therefore $n^{2}+2 n$ is divisible by $n$. Hence, for $n^{2}+2 n$ to be prime, $n$ can only have the value 1 .
When $n=1$, we have $n^{2}+2 n=3$, which is prime.
Therefore there is just one positive integer value of $n$ for which $n^{2}+2 n$ is prime.