My teacher once said, for any positive number n, nk−1 would always have n−1 as a factor for all positive odd values of k. Could anyone tell me the proof? I have written my approach below.
Assuming n−1 to be a factor.
nk−1=(n−1)x
nk=(n−1)x+1
k=lg((n−1)x+1)/lg(n)
If my approach is right could you tell me where should I go from here?
Edit : This clears my doubt.
Edit 2 : My question is "why for any positive number n, nk+1 would always have n+1 as a factor for all positive odd values of k ".
Answer
More generally, if a,b∈Z, n∈Z+, then a−b∣an−bn.
It follows from an−bn=(a−b)(an−1+an−2b+⋯+bn−1)
Edit: Since it seems you wanted to ask about the fact that for n∈Z with odd k∈Z+ we have n+1∣nk+1:
More generally, for a,b∈Z and odd n∈Z+ we have an+bn=(a+b)×
×(an−1−an−2b+an−3b2−⋯−abn−2+bn−1)
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