Find the maximum integer value $n$ such that $2^n\mid 3^{1024} -1$ Since $2^{10} = 1024$ then we can $3^{1024}-1 = ( 3 - 1 )\Pi_{i=0}^{9} {3^{2^i}+1}$
And then we can start eliminating the $3-1=2$ and $(3^2-1)= 2\times 5$
But then? I guess I could calculate, but this is not different as starting calculating in first place.
What is a better method?

Hint
For $i \geq 1$ you have:
$$3^{2^i}+1 \equiv (-1)^{2^i}+1 \equiv 1+1 \equiv 2 \pmod{4}$$
