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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 $$1<n<10000$$?