In the following proposition from Geiges book Introduction to Contact Topology he refers to "even" elements of $H^2(M;\mathbb{Z})$. What I am confused about is this: If I have a contact manifold of dimension n+1 and I know that $H^n(M;\mathbb{Z}) = H^n(M;\mathbb{Z}_2) = \mathbb{Z_2}$, can I conclude that the only possible Euler class for the contact structure, as an n-plane bundle, is zero in light of the below information?
It seems to me that this is the same as asking about the relationship between $H^k(M;\mathbb{Z}) / 2$ and $H^k(M;\mathbb{Z}_2)$