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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 $w(E) = 1$.

I read here (Prop 7): https://amathew.wordpress.com/2011/10/18/thoms-construction-of-the-stiefel-whitney-classes/#defeq

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.

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