Question

Consider the following statements:
If p is a prime such that $p+2$ is also a prime, then
1. $p(p+2)+1$ is a perfect square.
2. $12$ is a divisor of $p+(p+2)$ if $p>3$.
Which of the above statements is/are correct?

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ nor $2$

Answer 1

The correct option is C Both $1$ and $2$

If $p=3$, then $(p+2)=5$
$\therefore 3\times 5+1=16$ which is a perfect square.
So statement $1$ is true.
Again if $p=5$, then $p+2=7$ and $5,7$ are both prime.
Now $p+(p+2)=5+7=12$ and is divisible by $12$
Hence (c).