So, ${\lambda _{{{\max }_1}}}{T_1} = {\lambda _{{{\max }_2}}}{T_2}$

$ \Rightarrow {\lambda _0}T = \frac{{3{\lambda _0}}}{4}T' \Rightarrow \frac{{T'}}{T} = \frac{4}{3}$                       $...(i)$

According to $Stefan-Boltzmann\,law$, energy emitted unit time by a blck body is $Ae\sigma T',i.e.,$ power radiated,

$\therefore \,\,P \propto {T^4}$

$So,\frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{{T'}}{T}} \right)^4} \Rightarrow n = {\left( {\frac{4}{3}} \right)^4} = \frac{{256}}{{81}}$