# Problem 26

Solution to Problem 26. Solved in Python.

Problem: 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.

Notes: This is kind of a hackey solution, as there is no general method for finding reptend primes (source: http://mathworld.wolfram.com/FullReptendPrime.html). I didn’t feel like implementing it using some sort of string comparisons, because I preferred a purely mathematical solution (at least for this problem).

Other sources: