I think this is kind of the same track that Vetinari was on.

Qqy, your teacher must have been a sadist. That rule is true for all primes.

Theorem:
3 < p < +infinity
If p is prime, then either (p + 1) or (p - 1) is divisible by 6.

Proof: (Not my work)
(1) If p is prime, then it is odd. (p > 2)
(2) If p is odd, then (p + 1) and (p - 1) are even.
(3) If (p + 1) and (p - 1) are even, they are both divisible by 2.
(4) p-1, p, and p+1 are consecutive integers.
(5) Given any three consecutive integers, one of them must be divisible by three.
(6) p is not divisible by 3; it is prime.
(7) Either (p-1) or (p+1) must be divisible by 3. [(4), (5), & (6)]
(8) A number which is divisible by both 2 AND 3 is divisible by 6.
(9) One of (p-1) or (p+1) must be divisible by 6. [(3), (7), & (8)]