6 . Find the positive integer solutions to the equation $5(x y+y z+z x)=4 x y z$.

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6 . Find the positive integer solutions to the equation $5(x y+y z+z x)=4 x y z$.
6. The original equation can be transformed into $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{4}{5}$.

Assume $x \leqslant y \leqslant z$, then the original equation system has 12 sets of positive integer solutions: $(2,4,20),(2,20,4),(4$,
$$
\begin{array}{l}
2,20),(20,2,4),(4,20,2),(20,4,2),(2,5,10),(5,2,10),(10,2,5),(5, \\
10,2),(10,5,2),(2,10,5)
\end{array}
$$