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 $\displaystyle p$ must be odd. I would say $\displaystyle p\equiv1\mod{10}$ except when $\displaystyle p=5,$ because if $\displaystyle p\not\equiv1\mod{10}$ then one of $\displaystyle p,$ $\displaystyle p+2,$ $\displaystyle p+6,$ $\displaystyle p+8$ would be divisible by $\displaystyle 5.$ In the case $\displaystyle p=5,$ it is okay for $\displaystyle p$ to be divisible by $\displaystyle 5$ because $\displaystyle 5$ 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?