Reciprocal Cycles

Problem Statement

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.

Approach

The solution involves:

1. Implementing long division to find the recurring cycle length
2. Detecting when a remainder repeats (indicating the start of a cycle)
3. Testing all denominators d < 1000
4. Finding the one with the longest recurring cycle