Question

Tìm X nguyên để \(\frac{\sqrt{X}+1}{\sqrt{X}-1}\) có giá trị nguyên

Answer 1

Để \(\frac{\sqrt{X}+1}{\sqrt{X}-1}\) có giá trị nguyên

\(\Leftrightarrow\sqrt{X}+1⋮\sqrt{X}-1\)

\(\Leftrightarrow\sqrt{X}-1+2⋮\sqrt{X}-1\)

\(\Leftrightarrow2⋮\sqrt{X}-1\)(vì \(\sqrt{X}-1⋮\sqrt{X}-1\))

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{X}-1=1\\\sqrt{X}-1=2\\\sqrt{X}-1=-1\\\sqrt{X}-1=-2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\sqrt{X}=2\\\sqrt{X}=3\\\sqrt{X}=0\\\sqrt{X}=-1\left(loai\right)\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}X=4\\X=9\\X=0\end{matrix}\right.\)

Vậy \(X\in\left\{0;4;9\right\}\)