prove by induction $7 \mid 3^{3^n}+8$ Okay so ive been trying to prove this for about 5 hours... 
really need salvation from the geniouses around here.
prove  by induction
$7\mid 3^{3^n}+8$
i really need some directions on what to do here... 

First note that this is equivalent to $7$ dividing $3^{3^n}+1$. Clearly if $n=1$, then $3^{3^1}+1 = 28$ and $7$ divides this.
Let's suppose it's true for $n$. Then $3^{3^{n+1}} = 3^{3\cdot3^n} = (3^{3^n})^3$. We want then to show that $7$ divides $(3^{3^n})^3+1$. A nice factorization is that $a^3+b^3 = (a+b)(a^2-ab+b^2)$. So then we can write $(3^{3^n})^3+1$ as $(3^{3^n}+1)((3^{3^n})^2-3^{3^n}+1)$. Can you take it from here?
