Hola
The number of positive integers \[ x,y,z \] satisfying \[ x^{y^{z}}\cdot y^{z^{x}}\cdot z^{x^{y}}=5xyz \]
The equation is equivalent to:
\( x^{y^{z}-1}\cdot y^{z^{x}-1}\cdot z^{x^{y}-1}=5 \)
Since \( x,y,z \) are positive integers at least one of then must be \( 5 \). Suppose \( x=5 \). Then:
\( y^z-1=1\quad \Rightarrow{}\quad y^z=2\quad \Rightarrow{}\quad y=2,\quad z=1 \)
We obtain the solution \( (x,y,z)=(5,2,1) \). By the symmetry of the equation there are two other possibilities:
\( (x,y,z)=(1,5,2),(2,1,5) \)
Best regards.
