Question

Determinați elementele mulțimii
A={x aparține N* |
$\frac{ \sqrt{ {(2 \sqrt{3} - 4)}^{2} + \sqrt{28 + 16 \sqrt{3} } } }{x - 2}$
aparține Z}​

Answer 1

$\dfrac{\sqrt{(2\sqrt 3-4)^2}+\sqrt{28+16\sqrt{3}}}{x-2} = \dfrac{\sqrt{(2\sqrt 3-4)^2}+\sqrt{4(7+4\sqrt{3})}}{x-2} =\\ \\=\dfrac{\sqrt{(2\sqrt 3-4)^2}+2\sqrt{(\sqrt{3}+2)^2}}{x-2}=\dfrac{|2\sqrt 3-4|+2|\sqrt{3}+2|}{x-2}=\\ \\=\dfrac{4-2\sqrt{3}+2\sqrt{3}+4}{x-2} =\dfrac{8}{x-2}\in \mathbb{Z}\\\\\\\Rightarrow x-2 \in D_8 \Rightarrow x-2\in \{-8,-4,-2,-1,1,2,4,8\}\\ \\ \Rightarrow x\in \{-6,-2,0,1,3,4,6,10\}$

$\Rightarrow A = \{-6,-2,0,1,3,4,6,10\} \cap \mathbb{N}^*\\ \\ \Rightarrow \boxed{A =\{1,3,4,6,10\}}$