Solve-for-real-numbers-the-equation-x-2-3x-5-x-10-x-where-we-denoting-by-x-the-great-integer-part-of-x-

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

Solve for real numbers the equation

[\frac{x}{2}] + [\frac{3 x}{5}] = [\frac{x}{10}] + x, where we

denoting by [x] the great integer part

of x .

Answered by dumitrel last updated on 18/Aug/21

\Rightarrow x \in Z \Rightarrow x = 10 c + r, c \in Z, r \in {0, 1, \dots, 9}

\Rightarrow [5 c + \frac{r}{2}] + [6 c + \frac{3 r}{5}] = [c + \frac{r}{10}] + 10 c + r

5 c + [\frac{r}{2}] + 6 c + [\frac{3 r}{5}] = c + [\frac{r}{10}] + 10 c + r

[\frac{r}{2}] + [\frac{3 r}{5}] = [\frac{r}{10}] + r \Rightarrow r \notin {1, 3} \Rightarrow

x = 10 c + r, r \in {0, 2, 4, 5, 6, 7, 8, 9}

Commented by mathdanisur last updated on 18/Aug/21

nice S er thank you