What are the prime numbers $p$ such that $p+2$ and $p+4$ are also prime?
It is known that among three consecutive numbers, one of them will be a multiple of 3. The idea here is the same, and we will reason as we would for consecutive numbers. We know that $p$ can be written in the form $3k$, $3k+1$, or $3k+2$. Let's consider these three cases:

- If $p=3k$, $p$ is divisible by 3 and is prime, so $p=3$. Conversely, 3, 5, and 7 are prime.
- If $p=3k+1$, $p+2$ is divisible by 3, so $p+2=3$ and $p=1$, which is not prime.
- If $p=3k+2$, $p+4$ is divisible by 3, so $p+4=3$ and $p=-1$, which is not prime.

Thus, $p=3$ is the only value that works.