Every odd divisor of 2000 admits a solution

(that might contain negative terms).

For example, divide 2000 by 5 to get 400, the middle of five consecutive integers with sum 2000.

Check this: 398 + 399 + 400 + 401 + 402 = 2000

On the other hand, 2000/125 = 16, so 16 is the middle of a list of 125 consecutive integers with sum 2000. But many of these are negative.