#### Problem 26: Reciprocal Cycles

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. Find the value of *d*<1000 for which ^{1}/_{d} contains the longest recurring cycle in its decimal fraction part.

#### The Catch

How to find the length of a recurring cycle of ^{1}/_{d}

#### The Light

Use an algorithm which uses modulus operation to find the length of a recurring cycle:

- Take the modulus 1 % d. Call the result x.
- Times x by 10.
- Take the modulus x % d.
- Keep track of all found values for x.
- Repeat step 2 - 3 until x repeats again or x = 0.

For example, to find the recurring cycle length of 1/8:

1 % 8 = 1

10 % 8 = 2

20 % 8 = 4

40 % 8 = 0 (Stop the algorithm here; 1/8 has a recurring cycle length of 3)

#### The Code

import java.util.*; public class Problem26 { public static void main(String[] args) { int count = 1; int remainder = 0; int tmp = 10; int max = 0; int result = 0; for(int d = 2; d < 1000; d++) { ArrayList list = new ArrayList(); list.add(new Integer(1)); while(true) { remainder = tmp % d; if(list.contains(remainder) || remainder == 0) { if(count > max) { max = count; result = d; } count = 1; tmp = 10; break; } else { list.add(new Integer(remainder)); tmp = remainder * 10; count++; } } } System.out.println(result); } }