Question

Determinati solutiile intregi ale ecuatiei[x/10]=[x/8],unde [a] e partea intreaga a numarului a.

Answer 1

[tex][\frac{x}{10}]=[\frac{x}{8}]=^{not}r\\ Stim\ ca:\\ r\leq\frac{x}{8}\ \textless \ r+1\\ r\leq\frac{x}{10}\ \textless \ r+1\\ Daca\ inmultim\ a\ doua\ relatie\ cu\ -1\ obtinem:\\ -r-1\ \textless \ -\frac{x}{10}\leq-r\\ Prin\ adunarea\ celor\ doua\ relatii:\\ -1\ \textless \ \frac{x}{8}-\frac{x}{10}\ \textless \ 1\\ -1\ \textless \ \frac{x}{40}\ \textless \ 1|\cdot 40\\ -40\ \textless \ x\ \textless \ 40\Rightarrow \boxed{\bold{x\in (-40,40)}}[/tex]