# About the sum of the first half and the latter half of the cyclic numbers of a repeating decimal

Let us call the sum of the first half and the latter half of the cyclic numbers of an irreducible fraction 'a division sum' when the period of a repeating decimal is even. Also, let $\lambda(l)$ be the length of the repeating digits of $\frac 1l$ in decimal expansion.

Example 1 : The division sum of $\frac 17$ is $142+857=999$ and $\lambda(7)=6$ because $\frac 17=0.\dot 142\ 85\dot 7$.

Then, here is my question.

Question : Is the following true?

"Suppose that $p,q$ are distinct prime numbers other than $2$ or $5$. Then, either $(1)$ or $(2)$ holds.

$(1)$ $\lambda(p)=2^h\mu, \lambda(q)=2^h\nu\ (h\ge 1, \mu,\nu$ are odd$)\Rightarrow$ the division sum of $\frac{1}{pq}$ can be represented as $99\cdots 9$.

$(2)$ $\lambda(p)=2^e\mu, \lambda(q)=2^f\nu\ (0\le e\lt f, \mu,\nu$ are odd$)\Rightarrow$ the division sum of $\frac{1}{pq}$ is in the form of repeating the cyclic number of $\frac rp$ where $r$ is the minimum natural number such that $qr\equiv 2\$(mod $p).$"

Example 2 : The division sum of $\frac 1{77}=\frac{1}{7\cdot 11}=0.\dot 012\ 98\dot 7$ is $012+987=999$. Note that $\lambda(7)=6=2^1\cdot 3,\lambda(11)=2^1\cdot 1$.

Example 3 : The division sum of $\frac 1{21}=\frac{1}{3\cdot 7}=0.\dot 047\ 61\dot 9$ is $047+619=666$. Since we know that $\lambda(3)=1=2^0\cdot 1,\lambda(7)=6=2^1\cdot 3$, $7r\equiv 2\$(mod$3)$ leads $r=2$. Note that $\frac 23=0.\dot 6$.

Example 4 : The division sum of $\frac 1{949}=\frac{1}{13\cdot 73}=0.\dot 001053740779\ 76817702845\dot 1$ is $1053740779+768177028451=769230\ 769230$. Since we know that $\lambda(13)=6=2^1\cdot 3,\lambda(73)=8=2^3\cdot 1$, $73r\equiv 2\$(mod$13)$ leads $r=10$. Note that $\frac{10}{13}=0.\dot 76923\dot 0$.

Motivation : We know that a fraction in lowest terms with a prime denominator other than $2$ or $5$ (i.e. coprime to $10$) always produces a repeating decimal. We can prove that the division sum of $\frac 1p$ can be represented as $99\cdots 9$ when $p$ is a prime other than $2$ or $5$. I've been thinking about its generalization. Then, I reached the above expectation. I would like to know not only whether this expectation is true but also how to prove that with relevant references if possible.

Remark : This question has been asked previously on math.SE without receiving any answers.

By the way, if my expectation is true, the following follows.

"Suppose that $p\ge 7$ is a prime number and that the period of the repeating decimal $\frac{1}{3p}$ is even. If $p\equiv 1\ ($mod $3)$, then the division sum of $\frac{1}{3p}$ can be represented as $66\cdots 6$. If $p\equiv 2\ ($mod $3)$, then the division sum of $\frac{1}{3p}$ can be represented as $33\cdots 3.$"