Prove that $\{0^n 1^{n\cdot m} : n,m \in \mathbb{N}\}$ is not context-free This is a homework problem I have spent several hours on. A "hint" is given that we may use this fact: If $n,j,k \in \mathbb{N}$ satisfy $ n \geq 2$ and $1 \leq j+k \leq n$, then $n^2+j$ does not evenly divide $n^3+k$.

I cannot find any way to apply this fact. It leads me to believe I should use the string $0^{p^2}1^{p^3}$ or something like that, but I am really just not sure. The pumping lemma has given me trouble since the non regular language version.

Even small hints greatly appreciated at this point.
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.