Problem 417

Reciprocal cycles II

A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:

^{1}/_{2}= 0.5 ^{1}/_{3}= 0.(3) ^{1}/_{4}= 0.25 ^{1}/_{5}= 0.2 ^{1}/_{6}= 0.1(6) ^{1}/_{7}= 0.(142857) ^{1}/_{8}= 0.125 ^{1}/_{9}= 0.(1) ^{1}/_{10}= 0.1

Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that ^{1}/_{7} has a 6-digit recurring cycle.

Unit fractions whose denominator has no other prime factors than 2 and/or 5 are not considered to have a recurring cycle.

We define the length of the recurring cycle of those unit fractions as 0.

Let L(n) denote the length of the recurring cycle of 1/n. You are given that L(n) for 3 n 1 000 000 equals 55535191115.

Find L(n) for 3 n 100 000 000

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