[tex]\it [x]=3\{x\}+1\ \ \ \ \ (1) \\ \\ \ [x\ ] \in \mathbb{Z} \stackrel{(1)}{\Longrightarrow}\ 3\{x\} =k\in\mathbb{Z} \Rightarrow \{x\}=\dfrac{k}{3}\ \ \ \ \ \ (2)\\ \\ x=[x]+\{x\}\ \stackrel{(1),(2)}{\Longrightarrow}\ x=k+1+\dfrac{k}{3} \Rightarrow x=\dfrac{4k+3}{3},\ k\in\mathbb{Z}[/tex]
