Kernel of multiplication-by-n map between abelian varieties

Question

My question is about Remark 7.3 from Milne's note on abelian varieties: https://www.jmilne.org/math/CourseNotes/av.html (ver. 2.0).

Let $A$ be an abelian variety over a field $k$; Let $A_n(k):=\ker([n]:A(k)\to A(k))$ be the kernel of the multiplication-by-$n$ map (contained in the $k$-points of $A$). Assume $\text{char}(k)\nmid n$ so that the morphism $[n]:A\to A$ is finite étale.

Milne says then: If $k$ is separably closed, then $\#A_n(k)=\deg([n])$. I want to know

1. How to deduce the above claim.
2. An example where $\#A_n(k)\neq \deg([n])$ if we drop the assumption on $k$.