Divisor of fraction is divisor of numerator or denominator?

Question

If we have $$q \mid \frac{x^a – 1}{x – 1},$$

does that mean that $q$ is divisor of $x^a – 1$ or $q$ is divisor of $x-1$ ?

Answer 1

Divisibility is transitive. So if $a \mid \dfrac{b}{c}$, since $\dfrac{b}{c} \mid b$ (as $\dfrac{b}{c} \cdot c = b$) we have $a \mid b$.

Answer 2

$\frac{abcd}{ab} = cd$ since denominator $ab$ divides $abcd$ then $c|\frac{abcd}{ab}$. This means that $c$ is a divisor of the numerator.