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Question Number 151101 by mathdanisur last updated on 18/Aug/21

$$\mathrm{Solve}\mathrm{for}\mathrm{real}\mathrm{numbers}\mathrm{the}\mathrm{equation}$$$$\left[\frac{\mathrm{x}}{2}\right]+\left[\frac{3\mathrm{x}}{5}\right]=\left[\frac{\mathrm{x}}{10}\right]+\mathrm{x},\mathrm{where}\mathrm{we}$$$$\mathrm{denoting}\mathrm{by}\left[\mathbf{x}\right]\mathrm{the}\mathrm{great}\mathrm{integer}\mathrm{part}$$$$\mathrm{of}\mathbf{x}.$$

Answered by dumitrel last updated on 18/Aug/21

$$\Rightarrow x\in Z\Rightarrow x=10c+r,c\in Z,r\in \{0,1,...,9\}$$$$\Rightarrow [5c+\frac{r}{2}]+[6c+\frac{3r}{5}]=[c+\frac{r}{10}]+10c+r$$$$5c+\left[\frac{r}{2}\right]+6c+\left[\frac{3r}{5}\right]=c+\left[\frac{r}{10}\right]+10c+r$$$$\left[\frac{r}{2}\right]+\left[\frac{3r}{5}\right]=\left[\frac{r}{10}\right]+r\Rightarrow r\notin \{1,3\}\Rightarrow$$$$x=10c+r,r\in \{0,2,4,5,6,7,8,9\}$$$$$$

Commented by mathdanisur last updated on 18/Aug/21

$$\mathrm{nice}\mathbf{S}\mathrm{er}\mathrm{thank}\mathrm{you}$$

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