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How to prove that $x^a-1|x^b-1 \Longleftrightarrow a|b$.

Discussion in 'Mathematics' started by Cisplatin, Oct 8, 2018.

  1. Cisplatin

    Cisplatin Guest

    Prove that $x^a-1|x^b-1 \Longleftrightarrow a|b$, where $x \ge 2$ and $a,b,x \in \Bbb Z$.

    I've tried the following in attempting to solve this:

    $$a|b \rightarrow aq=b \rightarrow x^{aq}=x^b \rightarrow x^ax^q=x^b$$

    Because $x^q \in \Bbb Z$, it follows that $x^a|x^b$.

    This is as far as I have gotten; any help getting further is appreciated.

    Note: It may be that this identity is not true at all?

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