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.

Re: Find largest value d, with corresponding k, for following theorem – and proof!

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.

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

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