Counting vectors over finite fields (vector space)

Question

If $p$ is a prime,

1. Show that $\dfrac{\Bbb Z}{p^{n}\Bbb Z}$ is a vector space over $\dfrac{\Bbb Z}{p\Bbb Z}$ .

2. How many vectors are there in this vector space?

Answer 1

(1) For $\;n>1\;,\;\;\Bbb Z/p^n\Bbb Z\;$ is not a vector space over $\;\Bbb Z/p\Bbb Z\;$ , because if it were then for all $\;w\in\Bbb Z/p^n\Bbb Z\;$ we'd have that $\;0\cdot w=p\cdot w=0\;$, but this is impossible since $\;\Bbb Z/p^n\Bbb Z\;$ is a cyclic group of order $\;p^n\;$ ...

(2) Zero.