Suppose $M_n$ is a $n$ dimensional non-orientable manifold.

I am interesting in knowing whether the following statements are true:

- A characteristic class $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$ can be integrated on $M_n$ only when $p=2$.
- Suppose $n_1+n_2=n$, and $n_i, (i=1,2)$ are positive integers. $w_{n_i}^{(p)} \in H^{n_i}(M_n, \mathbb{Z}_p), i=1,2$. Then $w_{n_1}^{(p)}\cup w_{n_2}^{(p)}$ can be integrated on $M_n$ only when $p=2$.

If not true, is there a counter example?