## Reciprocal cycles II

### Problem 417

Published on Saturday, 2nd March 2013, 04:00 pm; Solved by 412; Difficulty rating: 35%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