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periodic cycle lengths

Description

Any rational number \(a/b\) can be written as a decimal expansion which either is finite or has a period, a recurring cycle. Looking at the reciprocal fractures \(1/n\) examples of such decimal expansions with a period are:

n decimal expansion period length
  \(3\)  \(0.\overline3\)  1
  \(7\)\(\)  \( 0.\overline{142857}\)  6
 \(12\) \( 0.08\overline{3}\)  1

Question

What is the sum of period lengths of the reciprocala \(1/n\) with \(1<n<10000\)?


What is the sum of period lengths of the reciprocala \(1/n\) with \(1<n<10000\)?