Let a complex number $\mathrm{\alpha },(\mathrm{\alpha }\ne 1)\text{ to be root of the equation }{\mathrm{Z}}^{\mathrm{p}+\mathrm{q}}-{\mathrm{Z}}^{\mathrm{p}}-{\mathrm{Z}}^{\mathrm{q}}+1=0$ where p, q are distinct primes. Then

${\mathrm{z}}^{\mathrm{p}+\mathrm{q}}-{\mathrm{z}}^{\mathrm{p}}-{\mathrm{z}}^2+1=0\Rightarrow \left({\mathrm{z}}^{\mathrm{p}}-1\right)\left({\mathrm{z}}^{\mathrm{q}}-1\right)=0$
${\mathrm{z}}^{\mathrm{p}}-1=0\text{ or }{\mathrm{z}}^{\mathrm{q}}-1=0$ 

$\text{ as }\alpha \text{ is root of (1), either }{\alpha }^p-1=0\text{ or }{\alpha }^q-1=0$

$\Rightarrow \text{ either }\frac{{\alpha }^p-1}{\alpha -1}=0\text{ or }\frac{{\alpha }^q-1}{\alpha -1}=0(\text{ as }\alpha \ne 1)$

$\Rightarrow \text{ either }1+\alpha +{\alpha }^2+\dots +{\alpha }^{p-1}=0$

$\text{ or }1+\alpha +\dots +{\alpha }^{q-1}=0$

$\text{ But }{\alpha }^p-1=0\text{ and }{\alpha }^{\mathrm{q}}-1=0\text{ cannot occur simultaneously as }\mathrm{p}\text{ and }\mathrm{q}\text{ are }$

$\text{ distinct primes, so neither }\mathrm{p}\text{ divides }\mathrm{q}\text{ nor }\mathrm{q}\text{ divides }\mathrm{p}\text{, }$