Consider the exact sequence $$ \cdots \rightarrow 0 \rightarrow \mathbb{Z} \overset{\times p}{\rightarrow} \color{blue} {\mathbb{Z}} \twoheadrightarrow \mathbb{Z} /p \mathbb{Z} \rightarrow 0 \rightarrow \cdots$$ Show that in the middle group $\color{blue} {\mathbb{Z}}$, the closed elements (or co-cycles) are the elements $p \mathbb Z$ and these are clearly the exact elements in this group.
My approach:
Multiplication map by $p$ takes $ x\in \mathbb Z$ to $p \mathbb Z$. Is this enough to say?