Divisor of fraction is divisor of numerator or denominator?

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 $x-1$ ?

• An example: $13$ divides $\frac{3^3 - 1}{3-1} = \frac{26}{2} = 13$. Who divides whom here? – pjs36 Jan 1 '17 at 22:00
• @TripleA Look at the usage guidance for divisors; it is not being used in the same sense as this question. Probably divisibility is more appropriate (and not modular arithmetic at all). – pjs36 Jan 1 '17 at 22:32

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 $a \mid b$.

• Does that mean that in the case: q|$\frac{x^a – 1}{x – 1}$ q "must" be divisor of $x^a-1$ ? – Math Newbie Jan 1 '17 at 22:45
• That's exactly what it means. Just take $a=q$ and $b=x^a-1$. – Xam Jan 1 '17 at 22:58

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