Prove that $\{0^n 1^{n\cdot m} : n,m \in \mathbb{N}\}$ is not context-free
If you use the pumping lemma on the word $w=0^{p^2} 1^{p^3}$, consider the partition $w=xyzuv$, where $|yzu|\le n$ and $|yu|&gt;0$ ($n$ being the length of $w$). 
It is easy to prove that from all the cases (that is, from all the possibilities for $y$ and $u$), the only non-trivial case is when $y=0^i$ and $u=1^j$, in which case the hint you mentioned finishes the job.