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.
using Common.Math; using System; namespace ProjectEuler_26 { internal class Program { public static void Main(string[] args) { Tupleresult = new Tuple (0, 0); // start by getting primes up to 1000 var primes = Eratosthenes.GetPrimes(1000); // loop backwards for (int i = primes.Length - 1; i >= 0; i--) { int count; // we know that the most a cyclical decimal will repeat is d-1 if ((count = GetCyclicalCount(primes[i])) == primes[i] - 1) { result = new Tuple (count, primes[i]); break; } } Console.WriteLine($"The value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part is {result.Item2} with {result.Item1} numbers in the cycle."); } private static int GetCyclicalCount(int prime) { int period = 1; // using Fermat's Little Theorim you can get the count of numbers in the cycle referred to as a period // Fermat's Little Theorim is defined as a^p=a%p // where a is the number base and p is a prime of that number base // this means that the count of cyclical numbers is the first modular power where the period test is not 1 while (ModularPow(10, period, prime) != 1) { period++; } return period; } private static int ModularPow(int @base, int exponent, int modulus) { if (modulus == 1) return 0; int result = 1; for (int i = 1; i <= exponent; i++) { result = (result * @base) % modulus; } return result; } } }
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