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:

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

Where $0.1(6)$ means $0.166666\ldots$, and has a $1$-digit recurring cycle. It can be seen that $\frac{1}{7}$ has a $6$-digit recurring cycle.

Find the value of $d<1000$ for which $\frac{1}{d}$ 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：

这里括号表示循环节，如$0.1(6)$就是指$0.166666\ldots$，循环节的长度为$1$。可以看出，$\frac{1}{7}$的循环节长度为$6$。

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