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Find all positive Integers x , y ,z satisfying x^(y^z).y^(z^x).z^(x^y)=5xyz

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Solution:  Given ,

                     x^y^z.y^z^x.z^x^y=5xyz

             x,y,z are integers and 5 is a prime number and given 

       equation is 

  Rightarrow         x^y^z.y^z^x.z^x^y=5xyz

On dividing both sides of the equation by xyz

               x^y^z-1.y^z^x-1.z^x^y-1=5

So, the different possibilities  are 

              x^y^z-1=5hspace0.2cmor hspace0.1cm x^y^z-1=1 hspace0.2cmor hspace0.1cmx^y^z-1=1

             y^z^x-1=1hspace0.2cm y^z^x-1=5 hspace0.2cmy^z^x-1=1

          z^x^y-1=1hspace0.2cm z^x^y-1=5 hspace0.2cmz^x^y-1=1

 Taking first column

          x=5,y^z-1=1,y^z=2,y=2 hspace0.2cmand hspace0.1cmz=1

and these values are satisfying the other expression in first column

Similarly , from the second column , we get 

       y=5,z=2hspace0.2cmand hspace0.2cmx=1  from third  z=5 ,x=2 hspace0.2cmand hspace0.2cmy=1.

 

 

Posted by

Deependra Verma

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