Solution:
We know that $85^{9}-21^{9}$ has $85-21=64$ as a factor, and $6^{9}$ also has $64$ as a factor, so the sum is divisible by $64$.

Similarly, $-21^{9}+6^{9}$ is divisible by $-21+6=-15$, which means it is divisible by $5$. Since $85^{9}$ is also divisible by $5$, the whole sum is divisible by $5$.

Finally, $85^{9}+6^{9}$ is divisible by $85+6=91$, so it is divisible by $7$. Since $21^{9}$ is also divisible by $7$, the sum is divisible by $7$.

Since the sum is divisible by $64$, $5$, and $7$, it is also divisible by $64 \cdot 5 \cdot 7 = 2240$.