Collatz mapping on $\mathbb{Z}\mathrm{/}10\mathbb{Z}$

We introduce the \emph{Collatz conjecture} and its history. Some definition that this conjecture has, will be expressed and with these we try to explain some good lemma to justify the main properties of the \emph{Collatz conjecture}. With these lemmas a situation made to go through to write some mainly properties of Collatz graph on $\mathbb{Z}\mathrm{/}10\mathbb{Z}$. Then with something provided it will go to draw the graph on $\mathbb{Z}\mathrm{/}10\mathbb{Z}$.