Let $f:M\to N$ be a smooth map between compact connected oriented n-dimensional differentiable manifolds. Assume that the image of the map on fundamental groups $f_\pi:\pi_1(M, m_0)\to \pi_1(N,f(m_0))$ has finite index $D$. Prove that $\deg(f)$ is divisible by $D$.
I just have no idea how something about the fundamental groups can reveal information about the degree of a map! It may be far beyond my knowledge.
Let $F = f_{\pi}(\pi_1(M))$. The classification of covering spaces tells us that there is a covering space $p: X \to N$ such that $p_{\pi}(\pi_1(X)) = F$. In particular, $f$ has a unique lifting $\tilde{f}$ such that $f = p\circ \tilde{f}$. Note that for any finite sheeted covering, the degree of the map is exactly the number of sheets, so the degree of our map $p$ is exactly the index of $F$ in $\pi_1(N)$. Then since the degree is multiplicative, we have that
$$\text{deg}(f) =\left(\text{deg}(\tilde{f})\right) \left(\text{deg}(p)\right) $$
And by the above comment the result.