Without a calculator, find a factor $85^{9}-21^{9}+6^{9}$ that is between 2000 and 3000 .
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 $64 \cdot 5 \cdot 7=2240$.