If $L_1,\,L_1L_2$ are regular languages with $L_1\neq \emptyset,\,\lambda\notin L_1,\,\lambda\notin L_2,$ is $L_2$ regular?
How about this? $L_1=\{a^n\mid n&gt;0\}$, $L_2 = \{a^{m^2}\mid m&gt;0\}$. Certainly $L_1$ and $L_1L_2$ are regular, but $L_2$ certainly isn't.