---
id: 5900f50d1000cf542c51001f
title: 'Problem 417: Reciprocal cycles II'
challengeType: 5
forumTopicId: 302086
dashedName: problem-417-reciprocal-cycles-ii
---

# --description--

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

# --hints--

`euler417()` should return 446572970925740.

```js
assert.strictEqual(euler417(), 446572970925740);
```

# --seed--

## --seed-contents--

```js
function euler417() {

  return true;
}

euler417();
```

# --solutions--

```js
// solution required
```