Is the language $\{a^n b^m : 1000|nm \}$ regular?
Notice that $1000 = 2^3 5^3$. Then $1000 | nm$ iff for some $0 \le i,j \le 3$,
$n$ is multiple of $2^i 5^j$ and $m$ is a multiple of $2^{3-i} 5^{3-j}$ (notice that there are only finitely many choices for $i$ and $j$).
Then $L$ can be obtained as the union of the languages $L_{i,j} = \{a^n b^m : 2^i 5^j|n \;\mbox{ and }\; 2^{3-i} 5^{3-j}|m\}$.
Each of these languages  $L_{i,j}$ is the concatenation of $L'_{i,j} = \{ a^n : 2^i 5^j|n\}$ with $L''_{i,j} = \{b^m : 2^{3-i}5^{3-j}|m\}$.
It is easy to see that each $L'_{i,j}$ and each $L''_{i,j}$ is regular. By the closure properties of regular languages $L$ is regular.