A problem by Kwesi Levy

Level pending

Let \(p\) be any positive prime number. What is the length of the longest repetend* required to express \(0.1_{10}\) in base-\(p\)?

*The "repetend" is the repeating block of a non-terminating, rational decimal. For example, in base-10, \(\frac{1}{3} = 0.\overline{3}\) has a repetend of length 1 while \(\frac{1}{990} = 0.0\overline{01}\) has a repetend of length 2 (we ignore the leading decimal[s]).

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