In general, for any given $a_{1}, \cdots, a_{k}$, there must exist $c$, such that
$$\delta_{m}(c)=\left[\delta_{m}\left(a_{1}\right), \cdots, \delta_{m}\left(a_{k}\right)\right]$$

In particular, if $a_{1}, \cdots, a_{k}$ is a set of reduced residues modulo $m$, then the index $\delta_{m}(c)$ of $c$ in the above equation is the maximum index modulo $m$.