[tex] \frac{5}{2 - x} \: \in \: \mathbb{Z} = > (2 - x) \: \in \: D_{5}[/tex]
[tex]D_{5}=\left \{ \pm1,\pm5 \right \}[/tex]
[tex]1)2 - x = - 1[/tex]
[tex] - x = - 1 - 2[/tex]
[tex] - x = - 3 \: | \times ( - 1)[/tex]
[tex]x = 3 \: \in \: \mathbb{Z}[/tex]
[tex] \frac{5}{2 - 3} = \frac{5}{ - 1} = - 5 \: \in \: \mathbb{Z}[/tex]
[tex]2)2 - x = 1[/tex]
[tex] - x = 1 - 2[/tex]
[tex] - x = - 1 \: | \times ( - 1)[/tex]
[tex]x = 1 \: \in \: \mathbb{Z}[/tex]
[tex] \frac{5}{2 - 1} = \frac{5}{1} = 5 \: \in \: \mathbb{Z}[/tex]
[tex]3)2 - x = - 5[/tex]
[tex] - x = - 5 - 2[/tex]
[tex] - x = - 7 \: | \times ( - 1)[/tex]
[tex]x = 7 \: \in \: \mathbb{Z}[/tex]
[tex] \frac{5}{2 - 7} = \frac{5}{ - 5} = - 1 \: \in \: \mathbb{Z}[/tex]
[tex]4) 2 - x = 5[/tex]
[tex] - x = 5 - 2[/tex]
[tex] - x = 3 \: | \times ( - 1)[/tex]
[tex]x = - 3 \: \in \: \mathbb{Z}[/tex]
[tex] \frac{5}{2 - ( - 3)} = \frac{5}{2 + 3} = \frac{5}{5} = 1 \: \in \: \mathbb{Z}[/tex]
