What a Proof Is · Superchallenge problem

A proof reads: "for odd $n$, $8 \mid n^2 - 1$; since $n^4 - 1 = (n^2 - 1)(n^2 + 1)$ and $n^2 + 1$ is even, $16 \mid n^4 - 1$; continuing the same way, $\ldots$" What is the largest power of $2$ that divides $n^{16} - 1$ for EVERY odd $n$? (Give the power itself, such as $16$.)