When is $\frac {t^a - 1} {t^b -1} $ an integer?

Question

Given three positive integers $t$, $a$ and $b$, I'm interested to know when the fraction $$\frac {t^a - 1} {t^b -1} $$ is an integer itself. Excluding any trivial cases, by trying out different values I've come to the following conjecture: $$\frac {t^a - 1} {t^b -1} \in \mathbb{Z} \iff a\equiv 0\pmod b $$ However I haven't had any luck proving this or finding any counter-examples. Any hint is appreciated. Thank you.

Answer 1

Fix $b$ and $t$ and consider the question of which $a$ are such that $\frac{t^a - 1}{t^b - 1}$ is an integer; i.e., such that $t^a \equiv 1 \pmod{t^b - 1}$. Clearly, the order of $t$ within the multiplicative group of (invertible) integers modulo $t^b - 1$ is $b$ (as every smaller positive power of $t$ is strictly between $1$ and $(t^b - 1) + 1$). Thus, the answer to our question is those $a$ which are multiples of $b$, just as you conjectured.