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 $0.166666\ldots$, 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：

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

其中，括号表示循环节，如$0.1(6)$就是指$0.166666\ldots$，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$d$。