Suppose the potential energy between electron and proton at a distance $r$ is given by $-\frac{K{e}^2}{3r^3}.$ As per Bohr's quantization rule, the speed of electron in $n^{\mathrm{th}}$ orbit of $H$-atom is

Consider a hydrogen like atoms which are in $n^{\mathrm{th}}$ excited state and their energy is given by 

$E_n=\frac{-13.6}{n^2}{Z}^2$

In the emission spectra of this atom the most energetic photons have energy, $E_{\max }=117.5 \mathrm{eV}$ and least energetic photon have energy, $E_{\min }=2.75 \mathrm{eV}.$ Find the atomic number of the atom.