1.
D nguyên \(\Rightarrow\sqrt{x}+1\inƯ\left(3\right)\)
Mà \(\sqrt{x}+1\ge1;\forall x\ge0\)
\(\Rightarrow\sqrt{x}+1\in\left\{1;3\right\}\)
\(\Rightarrow\sqrt{x}\in\left\{0;2\right\}\)
\(\Rightarrow x\in\left\{0;4\right\}\)
2.
F nguyên \(\Rightarrow2\sqrt{x}-1\inƯ\left(5\right)\)
mà \(2\sqrt{x}-1\ge-1;\forall x\ge0\)
\(\Rightarrow2\sqrt{x}-1\in\left\{-1;1;5\right\}\)
\(\Rightarrow\sqrt{x}\in\left\{0;1;3\right\}\)
\(\Rightarrow x\in\left\{0;1;9\right\}\)
3.
\(Q=\dfrac{\sqrt{x}-3}{\sqrt{x}+2}=\dfrac{\sqrt{x}+2-5}{\sqrt{x}+2}=1-\dfrac{5}{\sqrt{x}+2}\)
Q nguyên \(\Rightarrow\dfrac{5}{\sqrt{x}+2}\) nguyên
\(\Rightarrow\sqrt{x}+2\inƯ\left(5\right)\)
Mà \(\sqrt{x}+2\ge2;\forall x\ge0\)
\(\Rightarrow\sqrt{x}+2=5\)
\(\Rightarrow\sqrt{x}=3\)
\(\Rightarrow x=9\)
4.
\(H=\dfrac{5-2\sqrt{x}}{\sqrt{x}-1}=\dfrac{3-2\left(\sqrt{x}-1\right)}{\sqrt{x}-1}=\dfrac{3}{\sqrt{x}-1}-2\)
Do 2 nguyên nên H nguyên khi \(\dfrac{3}{\sqrt{x}-1}\) nguyên
\(\Rightarrow\sqrt{x}-1\inƯ\left(3\right)\)
Mà \(\sqrt{x}-1\ge-1;\forall x\ge0\)
\(\Rightarrow\sqrt{x}-1\in\left\{-1;1;3\right\}\)
\(\Rightarrow\sqrt{x}\in\left\{0;2;4\right\}\)
\(\Rightarrow x\in\left\{0;4;16\right\}\)
5.
\(I=\dfrac{2\sqrt{x}-3}{2\sqrt{x}+1}=\dfrac{2\sqrt{x}+1-4}{2\sqrt{x}+1}=1-\dfrac{4}{2\sqrt{x}+1}\)
Do 1 nguyên nên I nguyên khi \(\dfrac{4}{2\sqrt{x}+1}\) nguyên
\(\Rightarrow2\sqrt{x}+1\inƯ\left(4\right)\)
Mà \(2\sqrt{x}+1\ge1;\forall x\ge0\)
Đồng thời \(2\sqrt{x}+1\) lẻ
\(\Rightarrow2\sqrt{x}+1\) là ước dương lẻ của 4
\(\Rightarrow2\sqrt{x}+1=1\)
\(\Rightarrow2\sqrt{x}=0\)
\(\Rightarrow x=0\)
6.
\(K=\dfrac{3-\sqrt{x}}{\sqrt{x}-1}=\dfrac{2-\left(\sqrt{x}-1\right)}{\sqrt{x}-1}=\dfrac{2}{\sqrt{x}-1}-1\)
Do 1 nguyên nên K nguyên khi \(\dfrac{2}{\sqrt{x}-1}\) nguyên
\(\Rightarrow\sqrt{x}-1\inƯ\left(2\right)\)
Mà \(\sqrt{x}-1\ge-1;\forall x\ge0\)
\(\Rightarrow\sqrt{x}-1\in\left\{-1;1;2\right\}\)
\(\Rightarrow\sqrt{x}\in\left\{0;2;3\right\}\)
\(\Rightarrow x\in\left\{0;4;9\right\}\)
