[tex]\frac{1}{x}+\frac{1}{0,(x)}+\frac{1}{0,0(x)}+\frac{1}{0,00(x)}=[/tex]
[tex]\frac{1}{x}+\frac{1}{\frac{x}{9}}+\frac{1}{\frac{x}{90}}+\frac{1}{\frac{x}{900}}=[/tex]
[tex]\frac{1}{x}+\frac{9}{x}+\frac{90}{x}+\frac{900}{x}=\frac{1000}{x}[/tex]
[tex]\frac{1000}{x}\in\mathbb{N}\Rightarrow x\mid1000[/tex]
[tex]x~este~cifra\Rightarrow x\in\{1,2,4,5,8\}[/tex]
[tex]1\cdot2\cdot4\cdot5\cdot8=\boxed{320}[/tex]
