Let $a$ be an integer such that $a>1$, $n\in\mathbb{N}$ and define the number $x_{m}$ by
$$x_{m}=\sum_{d|m}\mu(d)(-1)^{n}[a^{\frac{m/d}{(m/d,n)}}-1]^{(m/d,n)}$$
I must to proof that $x_{m}\neq 0, \text{for all} \ m$
My Approach If $m$ is a prime number it is easy to see that:
$$x_{m}=(-1)^n[(a-1)^m-(a-1)], \ \text{if} \ m|n$$
and
$$x_{m}=(-1)^n[a^m-a], \ \text{if} \ m \ \text{not divide} \ n $$
and in both cases it is obvious that $x_{m}\neq 0$
Does anybody has a hint for the general case?