In Milnor-Stasheff's Characteristic Classes, one of the problems says that the smallest nonzero $w_i(E)$ happens when $i$ is a power of 2 in the case where $w(E) = 1$.
That the first nonzero stiefel-whitney class is always given by a power of two. Is this true or does it only happen when the total sw class is 1? Where can I find a clear explanation?
If not, I am wondering if there are any conditions on manifolds that would tell us when $w(E) = 1$ or not. In particular, I am interested in closed, orientable manifolds.