The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?



Cyclic order relation in Zn
**Answer.** Under the condition
$$
\left\lbrace\frac{x(b-a)}n\right\rbrace<\left\lbrace\frac{x(c-a)}n\right\rbrace
\qquad\qquad(\*)
$$
where $\lbrace\cdot\rbrace$ denotes the fractional part of a real number.


The condition $R(a,b,c)$ is equivalent to $R(0,b-a,c-a)$ and means that the least residues of $b-a$ and $c-a$ modulo $n$ are ordered in the increasing order as integers. Note that the least residue of an integer $m$ modulo $n$ can be expressed by means of $n\cdot\lbrace m/n\rbrace$.


The relations ($\*$) are hardly related for different values of $x=1,2,\dots$: one can always construct $n$ and residues $a$ and $b$ in such a way that for a given range of $x\in X\_1\cup X\_2$ ($X\_1$ and $X\_2$ are two finite disjoint sets of $\mathbb Z\_{>0}$) the inequality ($\*$) hold for $x\in X\_1$ and does not hold for $x\in X\_2$.


