Question

aflati E, unde E={ x ∈ Z | $\frac{x+1}{x-5}$ ∈ Z, cu x par }

Answer 1

[tex]\it \dfrac{x+1}{x-5} \in \mathbb{Z} \Rightarrow x-5|x+1\ \ \ \ (1) \\\;\\ Dar,\ \ x-5|x-5\ \ \ \ (2)[/tex]

[tex]\it Din\ \ (1),\ (2) \Rightarrow x-5|x+1-(x-5) \Rightarrow x-5|x+1-x+5 \Rightarrow \\\;\\ \Rightarrow x-5|6 \Rightarrow x-5 \in D_6 \Rightarrow x-5 \in \{\pm1,\ \pm2,\ \pm3,\ \pm6\} \Rightarrow \\\;\\ \Rightarrow x-5 \in \{ -6,\ -3,\ -2,\ -1,\ 1, \ 2\ , 3, \ 6\}|_{+5} \Rightarrow \\\;\\ \Rightarrow x\in\{-1, 2, 3, 4, 6, 7, 8, 11 \} \\\;\\ Dar,\ \  x = par \Rightarrow E = \{2, 4, 6, 8\}[/tex]