Abstract: Orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: k-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is 2^{k+1}m for some integer m. Here we call a manifold k-orientable if the i^{th} Stiefel-Whitney class vanishes for all 0