Question

Find all positive integers $x,y,z$ satisfying $x^{y^z}.y^{z^x}.z^{x^y}=5xyz$.

• A. $1,5,2$
• B. $1,2,5$
• C. $2,1,5$
• D. $5,2,1$

Answer 1

Answer: (A), (C), (D)

The correct options are
A $1,5,2$
C $2,1,5$
D $5,2,1$

Here, $x,y,z$ are integers and $5$ is a prime number.
$\Rightarrow$ $\frac{x^{y^z}.y^{z^x}.z^{x^y}}{xyz}=5$
$\Rightarrow$ $x^{y^z-1}.y^{z^x-1}.z^{x^y-1}=5$ is only possible, if
Case I: $x^{y^z-1}=5$, $y^{z^x-1}=1$, $z^{x^y-1}=1$
$\Rightarrow$ $x=5$, $y^z-1=1$ $\Rightarrow$ $y^z=2$
$\Rightarrow$ $y=2$ and $z=1$
$\therefore$ $(x,y,z)=(5,2,1)$
Case II: $x^{y^z-1}=1$, $y^{z^x-1}=5$, $z^{x^y-1}=1$
$\therefore$ $(x,y,z)=(1,5,2)$
Case III: $x^{y^z-1}=1$, $y^{z^x-1}=1$, $z^{x^y-1}=5$
$\therefore$ $(x,y,z)=(2,1,5)$