Đáp án:

\(\left[ \begin{array}{l}
x = 36\\
x = 9\\
x = 4\\
x = 1
\end{array} \right.\)

Giải thích các bước giải:

\(\begin{array}{l}
M = \dfrac{{6\left( {\sqrt x  - 4 + 2} \right)}}{{\sqrt x \left( {\sqrt x  - 4} \right)}}\\
 = \dfrac{{6\left( {\sqrt x  - 4} \right) + 12}}{{\sqrt x \left( {\sqrt x  - 4} \right)}}\\
 = \dfrac{6}{{\sqrt x }} + \dfrac{{12}}{{x - 4\sqrt x }}\\
M \in Z\\
 \Leftrightarrow \left\{ \begin{array}{l}
\dfrac{6}{{\sqrt x }} \in Z\\
\dfrac{{12}}{{x - 4\sqrt x }} \in Z
\end{array} \right.\\
 \to \left\{ \begin{array}{l}
\left[ \begin{array}{l}
\sqrt x  = 6\\
\sqrt x  = 3\\
\sqrt x  = 2\\
\sqrt x  = 1
\end{array} \right.\\
\left[ \begin{array}{l}
x - 4\sqrt x  = 12\\
x - 4\sqrt x  =  - 12\left( {vô nghiệm} \right)\\
x - 4\sqrt x  = 6\\
x - 4\sqrt x  =  - 6\left( {vô nghiệm} \right)\\
x - 4\sqrt x  = 4\\
x - 4\sqrt x  =  - 4\\
x - 4\sqrt x  = 3\\
x - 4\sqrt x  =  - 3\\
x - 4\sqrt x  = 2\\
x - 4\sqrt x  =  - 2\\
x - 4\sqrt x  = 1\\
x - 4\sqrt x  =  - 1
\end{array} \right.
\end{array} \right. \to \left\{ \begin{array}{l}
\left[ \begin{array}{l}
x = 36\\
x = 9\\
x = 4\\
x = 1
\end{array} \right.\\
\left[ \begin{array}{l}
\sqrt x  = 6\\
\sqrt x  =  - 2\left( l \right)\\
\sqrt x  = 2 + \sqrt {10} \left( l \right)\\
\sqrt x  = 2 - \sqrt {10} \left( l \right)\\
\sqrt x  = 2 + 2\sqrt 2 \left( l \right)\\
\sqrt x  = 2 - 2\sqrt 2 \left( l \right)\\
\sqrt x  = 2\\
\sqrt x  = 2 + \sqrt 7 \left( l \right)\\
\sqrt x  = 2 - \sqrt 7 \left( l \right)\\
\sqrt x  = 3\\
\sqrt x  = 1\\
\sqrt x  = 2 + \sqrt 6 \left( l \right)\\
\sqrt x  = 2 - \sqrt 6 \left( l \right)\\
\sqrt x  = 2 + \sqrt 2 \left( l \right)\\
\sqrt x  = 2 - \sqrt 2 \left( l \right)\\
\sqrt x  = 2 + \sqrt 5 \left( l \right)\\
\sqrt x  = 2 - \sqrt 5 \left( l \right)\\
\sqrt x  = 2 + \sqrt 3 \left( l \right)\\
\sqrt x  = 2 - \sqrt 3 \left( l \right)
\end{array} \right.
\end{array} \right.\\
 \to \left[ \begin{array}{l}
x = 36\\
x = 9\\
x = 4\\
x = 1
\end{array} \right.
\end{array}\)