Proving divisibility for $256 \mid 7^{2n} + 208n - 1$ I can't come up with a way of proving this:
$$256 \mid 7^{2n} + 208n - 1\\ \forall n \in \Bbb N$$
I've tried by induction but couldn't see when to apply the inductive hypothesis...
$$P(n+1) = 7^{2n+2}+208n+207$$
How can I continue?
Thank you

We have transforming step by step suitably,
$$7^{2n}+208n-1=$$
$$(48+1)^n+(256-48)n-1=$$
$$(2^4\cdot3+1)^n+(256-2^4\cdot3)n-1 =$$
$$\sum_{k=0}^{k=n-1}\binom nk (2^4\cdot3)^{n-k}+1+256n-2^4\cdot3n-1=$$ 
$$=\sum_{k=0}^{k=n-2}\binom nk (2^4\cdot3)^{n-k}+256n+2^4\cdot3n-2^4\cdot3n=$$
$$=\sum_{k=0}^{k=n-2}\binom nk (2^4\cdot3)^{n-k}+256n\equiv 0\pmod{256}$$
