Question

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

Answer 1

$\it \dfrac{2x+1}{x-1} \in\mathbb{Z} \Rightarrow x-1 | 2x+1 \ \ \ \ \ \ (1)\\ \\ \\ Dar,\ x-1|x-1 \Rightarrow x-1|(x-1)\cdot2 \Rightarrow x-1|2x-2 \ \ \ \ \ (2)\\ \\ \\ (1),\ (2) \Rightarrow x-1|2x+1-(2x-2) \Rightarrow x-1|2x+1-2x+2 \Rightarrow x-1|3\Rightarrow\\ \\ \\ \Rightarrow x-1\in\{-3,\ -1,\ 1,\ 3\}|_{+1} \Rightarrow x\in \{-2,\ 0,\ 2,\ 4\}\Rightarrow A=\{-2,\ 0,\ 2,\ 4\}$