Help with completion of a simple proof by induction.

Question

I need some help completing the following proof by induction. I am not sure how to finish it.

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Thm. $\forall n \in \mathbb N (x-1)|(x^n-1)$

Pf by induction:

Inductive Hypothesis: Let $P(n) = (x-1)|(x^n-1)$

Base Case: n=1 $\frac{x-1}{x^1-1} = 1 \checkmark$

Inductive Step:

For $n \geq 1$ show that $P(n) \to P(n+1)$ is true.

Assume $P(n)$ is true, assume $(x-1)|(x^n -1)$

Examine $(x^{n+1} -1) \equiv x^n \cdot x^1 - 1 $

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What next? Not exactly sure how to finish this proof. Can someone take me through how to complete this?

Answer 1

$a\mid b$ if and only if $a\neq 0$ and $b=ak$ for some $k$.

Base case is clear, $(x-1)=(x-1)\cdot 1 \,\rightarrow\, (x-1)\mid (x-1)$.

Suppose that $(x-1)\mid(x^n-1)$.

Goal is to show that $(x-1)\mid (x^{n+1}-1)$, meaning just show $x^{n+1}-1=(x-1)k$ for some $k$.

$$x^{n+1}-1=x^{n+1}-x^n+x^n-1=x^n(x-1)+x^n-1$$ Apply the induction hypothesis and you finish the proof.

Answer 2

If You have proven that $x-1|x^{n}-1$, $x^{n+1}-1=x(x^{n}-1)+x-1$.

So it is proven by induction.