[tex]A=\left \{ n\:\in\:\mathbb{N} \:|\: \frac{12}{n + 3} \in\mathbb{N}\right \}[/tex]
[tex] \frac{12}{n + 3} \in\mathbb{N} = > n + 3\in \: D_{12} [/tex]
[tex]D_{12}=\left \{\pm 1,\pm2,\pm3,\pm4,\pm6,\pm12 \right \}[/tex]
[tex]1)n + 3 = 1 = > n = 1 - 3 = - 2\notin \: \mathbb{N}[/tex]
[tex]2)n + 3 = -1 = > n = -1 - 3 = - 4\notin \: \mathbb{N}[/tex]
[tex]3)n + 3 = 2 = > n = 2 - 3 = -1\notin \: \mathbb{N}[/tex]
[tex]4)n + 3 = -2 = > n = -2 - 3 = -5\notin \: \mathbb{N}[/tex]
[tex]5)n + 3 = 3 = > n = 3 - 3 = 0\in \: \mathbb{N}[/tex]
[tex]6)n + 3 = -3 = > n = -3 - 3 = -6\notin \: \mathbb{N}[/tex]
[tex]7)n+3=4=>n=4-3=1\in\mathbb{N}[/tex]
[tex]8)n+3=-4=>n=-4-3=-7\notin\mathbb{N}[/tex]
[tex]9)n+3=6=>n=6-3=3\in\mathbb{N}[/tex]
[tex]10)n+3=-6=>n=-6-3=-9\notin\mathbb{N}[/tex]
[tex]11)n+3=12=>n=12-3=9\in\mathbb{N}[/tex]
[tex]12)n+3=-12=>n=-12-3=-15\notin\mathbb{N}[/tex]
[tex]n\:\in\:\left \{ 0,1,3,9 \right \}[/tex]
[tex]=>A=\:\in\:\left \{ 0,1,3,9 \right \}[/tex]
