Here is a proof using the binomial theorem.





Every term of the binomial expansion of (8-1)2n
except the last two terms contain a factor of 8 to a power
2 or greater and thus they are all divisible by 64.  So the
above becomes upon writing out the first two and last two
terms of the binomial expansion: 



Simplifying and using the property of combinations C(2n,2n-1)=C(2n,1)=2n







The last two terms in the binomial expansion cancel with the 
16n-1, so all the remaining terms are divisible by 64.  Thus 
the theorem is proved.

Edwin