Excel in math, science, and engineering

New user? Sign up

Existing user? Sign in

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 7 months, 4 weeks ago

Sort by:

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 · 7 months, 3 weeks ago

Log in to reply

@Wen Z – Thank you so much – Sattik Biswas · 7 months, 3 weeks ago

@Wen Z – I mean \(\text{mod (n-1)}\) – Wen Z · 7 months, 3 weeks ago

Could you please provide a solution? – Sattik Biswas · 7 months, 3 weeks ago

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

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 · 7 months, 3 weeks ago

Log in to reply

– Sattik Biswas · 7 months, 3 weeks ago

Thank you so muchLog in to reply

– Wen Z · 7 months, 3 weeks ago

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

Could you please provide a solution? – Sattik Biswas · 7 months, 3 weeks ago

Log in to reply