Thread: Find largest value d, with corresponding k for following theorem: And proof!

Find the largest value of "d" and the corresponding value of k, for which this is true:

"If all of p, p+2, p+6 and p+8 are prime, then p= k (mod d) except in one case"

Also, what is the exceptional value of p that does not satisfy the theorem? And prove the theorem is true in all other cases.

Obviously must be odd. I would say except when because if then one of would be divisible by In the case it is okay for to be divisible by because is prime.

Thanks Sylvia, any idea how you'd go about a proof of the theorem?