Find all positive prime numbers $p$ such that $p+2$ and $p+4$ are prime as well.

Hint. Show that for most prime numbers $p$, either $p+2$ or $p+4$ is divisible by 3 .
For $p=3, p+2=5, p+4=7$ and these are obviously prime. For $p>3$, we know that $p$ is not divisible by 3 . The remainder of $p$ when divided by 3 can be either 1 or 2 . If it is one, then $p+2$ is divisible by 3 , if it is 2 , then $p+4$ is divisible by 3 . Hence $p=3$ is the only solution.