# Thread: prove (a^n)-1 is divisible by a-1

1. ## prove (a^n)-1 is divisible by a-1

i need help on trying to prove this (a^n)-1 is divisible by a-1 for every natural number n and every integer a > 1

2. Induction works. Skipping all the formalities ...

Inductive hypothesis: Assume ${\color{blue}a^k - 1}$ is divisible by $a - 1$. It remains to show that $a^{k+1} - 1$ is also divisible by $a-1$.

But note that:
\begin{aligned}a^{k+1} - 1 & = a^ka^1 - 1 \\ & = a^ka {\color{red} - a + a} - 1 \qquad (\text{We're adding '0'}) \\ & = a({\color{blue}a^k - 1}) + (a-1)\end{aligned}

And the conclusion follows

3. i'm still confused where does the a^1 come from.

4. Are you familiar with this property: $a^{m+n} = a^ma^n$

Simply imagine $m = k$ and $n = 1$.

5. Would it help if is said
$x^n-1=(x-1)(x^{n-1}+x^{n-2}\dots x+1)$
for every natural number n.