[tex] \frac{6}{1 - 2x} \: \in \: \mathbb{Z}[/tex]
[tex] = > 1 - 2x \: \in \: D_{6}[/tex]
[tex]D_{6} = \left \{ \pm1,\pm2,\pm3,\pm6 \right \}[/tex]
[tex]pt. \: x = - 1 [/tex]
[tex] \frac{6}{1 - 2 \times ( - 1)} = \frac{6}{1 + 2} = \frac{6}{3} = 2 \: \in \: \mathbb{Z}[/tex]
[tex]pt. \: x = 1[/tex]
[tex] \frac{6}{1 - 2 \times 1} = \frac{6}{1 - 2} = \frac{6}{ - 1} = - 6 \: \in \: \mathbb{Z}[/tex]
[tex]pt. \: x = - 2[/tex]
[tex] \frac{6}{1 - 2 \times ( - 2)} = \frac{6}{1 + 4} = \frac{6}{5} \notin \: \mathbb{Z}[/tex]
[tex]pt. \: x = 2[/tex]
[tex] \frac{6}{1 - 2 \times 2} = \frac{6}{1 - 4} = \frac{6}{ - 3} = - 2 \: \in \: \mathbb{Z}[/tex]
[tex]pt. \: x = - 3[/tex]
[tex] \frac{6}{1 - 2 \times ( - 3)} = \frac{6}{1 + 6} = \frac{6}{7} \: \notin \: \mathbb{Z}[/tex]
[tex]pt. \: x = 3[/tex]
[tex] \frac{6}{1 - 2 \times 3} = \frac{6}{1 - 6} = \frac{6}{ - 5} = - \frac{6}{5} \: \notin \: \mathbb{Z}[/tex]
[tex]pt. \: x = - 6[/tex]
[tex] \frac{6}{1 - 2 \times ( - 6)} = \frac{6}{1 + 12} = \frac{6}{13} \: \notin \: \mathbb{Z}[/tex]
[tex]pt. \: x = 6[/tex]
[tex] \frac{6}{1 - 2 \times 6} = \frac{6}{1 - 12} = \frac{6}{ - 11} = - \frac{6}{11} \: \notin \: \mathbb{Z}[/tex]
[tex] \frac{6}{1 - 2x} \: \in \: \mathbb{Z} = > x \: \in \:\left \{ -1,1,2 ,\right \}[/tex]
