$x+y+z=0$ $\Rightarrow y + z = -x$
$\Rightarrow ( y + z )^5 = -x^5$
$\Rightarrow y^5 + 5y^4z + 10y^3z^2 + 10y^2z^3 + 5yz^4 + z^5 + x^5 = 0$
$\Rightarrow x^5 + y^5 + z^5 + 5yz(y^3 + 2y^2z + 2yz^2 + z^3 ) = 0$
$\Rightarrow x^5 + y^5 + z^5 + 5yz [(y + z )( y^2 - yz + z^2 ) + 2yz( y+z) ] = 0$
$\Rightarrow x^5 + y^5 + z^5 + 5yz ( y + z )( y^2 + yz + x^2 ) = 0$
$\Rightarrow 2( x^5 + y^5 + z^5 ) - 5xyz (( y^2 + 2yz + z^2 ) + y^2 + z^2 ) = 0$
$\Rightarrow 2( x^5 + y^5 + z^5 ) = 5xyz( x^2 + y^2 + x^2 )$ $\Rightarrow \left ( dpcm \right )$