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:

$\begin{aligned} 1 / 2 & = 0.5 \\ 1 / 3 & = 0. (3) \\ 1 / 4 & = 0.25 \\ 1 / 5 & = 0.2 \\ 1 / 6 & = 0.1 (6) \\ 1 / 7 & = 0. (142857) \\ 1 / 8 & = 0.125 \\ 1 / 9 & = 0. (1) \\ 1 / 10 & = 0.1 \end{aligned}$

Where $0.1 (6)$ means $0.166666 \dots$ , 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.

倒数的循环节

单位分数指分子为 $1$ 的分数。分母为 $2$ 至 $10$ 的单位分数的十进制表示如下所示：

其中，括号表示循环节，如 $0.1 (6)$ 就是指 $0.166666 \dots$ ，循环节的长度为 $1$ 。可以看出， $1 / 7$ 的循环节长度为 $6$ 。

在所有满足 $d < 1000$ 的数中，求使得其倒数 $1 / d$ 的十进制表示中循环节最长的 $d$ 。