In the plots of multiplication graphs $\mu^n_m(k) = kn\ \%\ m$ (with $a\ \%\ b$ meaning $a$ modulo $b$) the dots are undeniably arranged along parallel lines with a well-defined common slope $\alpha$ given by

$$\tan(\alpha) = \frac{\mu^n_m(k_0) - n}{k_0 - 1}$$

with $k_0$ the number $k$ greater than $1$ which minimizes $(k - 1)^2 + (\mu^n_m(k) - n)^2$:

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[click image to enlarge]

These lines may have equal or different lengths – i.e. numbers of dots lying on them (possibly only two) – but in any case there is a specific number $\#L^n_m$ of them, e.g. $\#L^n_{50}=n$ for $n < 10$, $n \neq 8$ – or $\#L^{10}_{50}=5$.

My question is:

(How) can the number $\#L^n_m$ of these lines be given in a closed expression depending on $n$ and $m$?

The following expression at least gives correct results when all lines have equal length. In other cases, it gives numbers that are systematically too small, e.g. $\#L^{8}_{50}=6$ (instead of $7$) or $\#L^{9}_{50}=5$ (instead of $9$):

$$\#L^n_m= \begin{cases} k_0 - 1 & \text{if } |\tan(\alpha)| < 1\\ |\mu^n_m(k_0) - n| & \text{otherwise} \\ \end{cases}$$

or equivalently

$$\#L^n_m= \operatorname{max}\{k_0 - 1, |\mu^n_m(k_0) - n|\}$$

Is there a chance to solve this puzzle? (Its motivation will hopefully become clear in a follow-up question concerning quadratic residues modulo $m$, so please stay tuned.)


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