Prove that if $p, p+2$ and $p+4$ are primes, then $p=3$

Question

Since $p+2$ and $p + 4$ are prime, $3$ doesn't divide either of them. But then $3 \mid p+3$, which implies $3 \mid p$. Since $p$ is prime, then $p = 3$.

Is it this simple?

Answer 1

$3$ has to divide exactly one of $p, p+2, p+4$ (from simple congruency conditions). Since these are assumed to be primes, this implies $3$ IS one of $p, p+2, p+4$. Checking the $3$ conditions, this leads to $p=3$.

Answer 2

Hint $ $ They're $\,\equiv p,\, p\!+\!1,\, p\!+\!2\pmod 3\,$ so one is divisible by $3$