What are the integers $p$ such that $p, p+2$ and $p+4$ are all prime?
We look modulo 3:

If $p \equiv 0(\bmod 3):$ then $p=3$.

If $p \equiv 1(\bmod 3): p+2 \equiv 0(\bmod 3)$ and thus $3 \mid p+2$ which is prime i.e. $p+2=3:$ impossible.

If $p \equiv 2(\bmod 3): p+4 \equiv 0(\bmod 3)$ and thus $3 \mid p+4$ which is prime i.e. $p+4=3:$ impossible.

Conversely, $p=3$ works: 3, 5, and 7 are indeed prime.