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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 |

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 \(1<n<10000\)?