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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 5 months, 3 weeks 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 \(\text{mod n}\) – Wen Z · 5 months, 3 weeks ago

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@Wen Z – Thank you so much – Sattik Biswas · 5 months, 3 weeks ago

@Wen Z – I mean \(\text{mod (n-1)}\) – Wen Z · 5 months, 3 weeks ago

Could you please provide a solution? – Sattik Biswas · 5 months, 3 weeks ago

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## 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 \(\text{mod n}\) – Wen Z · 5 months, 3 weeks ago

Log in to reply

– Sattik Biswas · 5 months, 3 weeks ago

Thank you so muchLog in to reply

– Wen Z · 5 months, 3 weeks ago

I mean \(\text{mod (n-1)}\)Log in to reply

Could you please provide a solution? – Sattik Biswas · 5 months, 3 weeks ago

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