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Let \(n\geqslant 2\) and \(k\) be any positive integers. Prove that \((n-1)^2 | (n^k - 1) \) if and only if \((n-1) | k \).

Note by أحمد الحلاق 1 year, 10 months ago

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Just use the identity

\(\frac{n^k-1}{n-1}=n^{k-1}+\ldots+n^1+1\)

and take the resulting equation \(\text{mod n}\)

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Thank you so much

I mean \(\text{mod (n-1)}\)

Could you please provide a solution?

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Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestJust use the identity

\(\frac{n^k-1}{n-1}=n^{k-1}+\ldots+n^1+1\)

and take the resulting equation \(\text{mod n}\)

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Thank you so much

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I mean \(\text{mod (n-1)}\)

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Could you please provide a solution?

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