Question

Show that $p-1$ is a factor of $p^{10}-1$ and also $p^{11}-1$.

Answer 1

Given that $p-1$ is a factor of $p^{10}-1$ and also $p^{11}-1$

If $p-1$ is a factor then by factor theorem, $p-1=0$ or $p=1$

$\therefore f\left(p\right)=p^{10}-1$

$\Rightarrow f\left(1\right)=1^{10}-1=1-1=0$

$f\left(p\right)=p^{11}-1$

$\Rightarrow f\left(1\right)=1-1=0$

Hence $(p-1)$ is a factor.