In be-twin primes

If $p =3,$ then both $p$ and $p+2$ are prime numbers.

$p + 1 = 3 +1 = 4 ~ \text{is not divisible by 6}$

Though it is almost always true (see below), there is one counter-example : $3$ and $5$ are twin primes but $4$ is not divisible by $6$ .

In all other cases, it is because in a triplet of consecutive numbers $\{$ $p$ , $p+1$ , $p+2$ $\}$ , at least one is divisible by $2$ and one exactly is divisible by $3$ and because $p$ and $p+2$ are prime numbers, $p+1$ should be divisible by both $2$ and $3$ , hence by $6$ .