Question

Aflati elementele multimii A={x€Z| (2x+1/x-1)€Z}

Answer 1

$\displaystyle\\ A=\left\{x\in Z \Big| \frac{2x+1}{x-1} \in Z \right\}\\\\ \frac{2x+1}{x-1}=\frac{2x -2+2+1}{x-1}=\frac{2x -2+3}{x-1}=\\\\ =\frac{2x -2}{x-1}+\frac{3}{x-1}=\frac{2(x -1)}{x-1}+\frac{3}{x-1}=2+\frac{3}{x-1}\\\\ 2\in Z\\\\ \text{Ne ocupam de: }~~\frac{3}{x-1}\\\\ \frac{3}{x-1} \in Z\\\\ \Rightarrow~~(x-1) \in D_3\\\\ D_3= \{-3;~-1;~1;~3\}\\\\ x-1 = -3 ~=\!\!>~x = -3+1=-2\\ x-1 = -1 ~=\!\!>~x = -1+1=0\\ x-1 = 1 ~=\!\!>~x = 1+1=2\\ x-1 = 3 ~=\!\!>~x = 3+1=4\\\\ \boxed{A=\{-2;~0;~2;~4\}}$