Problem 26 was the first Project Euler problem I've yet encountered that I had to do research to solve. The problem asks for the integer under 1,000 whose reciprocal has the longest recurring cycle in its infinitely repeating decimal representation:

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.

This is the first problem in Project Euler to ask about non-integral values and it is designed to be unsolvable by floating-point arithmetic, as many terminating decimal values are repeating when represented as binary numerals: 1/10, for example, is 0.1 in base-10 but 0.00011 in binary. After some time reading basic number theory on Wolfram and Wikipedia, I found a few rules that could be used to find the answer:

• First, only denominators that are both prime and coprime with 10 should be evaluated. Non-prime denominators can resolve into a repeating decimal, but it will be preceded by a non-repeating sequence, and the length of the recurring cycle will be equal to that of its smallest prime factor coprime with 10. For example, 14 isn't prime, and 1/14 has a recurring cycle in its infinitely repeating decimal, but the length of that cycle is equal to the length of the recurring cycle in 1/7, i.e., the reciprocal of the smallest prime factor of 14 not coprime with 10.
• Second, the length of the recurring cycle of a repeating decimal 1/`x` is equal to `p` when 10`p` mod `x` = 1.
• Third, solving the problem by applying the above rule would involve manipulating huge numbers, i.e., BigIntegers

The large amount of BigInteger arithmetic in my code meant that it took a couple of seconds to execute, but it did work.

```using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace Problem026
{
class Program
{
static void Main(string[] args)
{
int ceiling = 1000;
int[] repeatingPeriod = new int[ceiling];
List<int> primesUnderCeiling = sieveOfEratosthenes(ceiling);
int hauptExponent = 0;
foreach (int i in primesUnderCeiling)
{
if (1000000 % i == 0)
{
repeatingPeriod[i] = 0;
}
else
{
do
{
hauptExponent++;
repeatingPeriod[i] = hauptExponent;
} while (BigIntModulo(10, hauptExponent, i) != 1);
hauptExponent = 0;

}
}

Console.WriteLine("Problem 26: {0}", Array.IndexOf(repeatingPeriod, repeatingPeriod.Max()));
}

private static int BigIntModulo(int p, int hauptExponent, int divisor)
{
System.Numerics.BigInteger power = p;
int modulus = 0;

for (int j = 1; j < hauptExponent; j++)
power *= 10;

modulus = (int)(power % divisor);

return modulus;
}

private static List<int> sieveOfEratosthenes(int ceil)
{
List<int> eratosthenes = new List<int>();
for (int i = 2; i < ceil; i++)