Show that $9|2^{2n+1}+2^{4n+1}-4$ 
Show that $9|2^{2n+1}+2^{4n+1}-4$ ( n is a positive integer):
  1-Using induction
  2-Don't use induction  

I have noticed that $2^m \equiv 2^{m+6} \pmod 9 $ but couldn't use it to solve the problem!

$2^{2n+1}+2^{4n+1}-4 = f(2^{2n})$ for $f(x)=2x^2+2x-4=2(x-1)(x+2)$.
Now $2^{2n} \bmod 9$ is $1,4,7,1,4,7,\dots$ and
$f(1) \equiv 0 \bmod 9$
$f(4) = 36 \equiv 0 \bmod 9$
$f(7) \equiv 0 \bmod 9$
So $f(2^{2n}) \equiv 0 \bmod 9$ for all $n$.
