Number of ways in which $1000$ can be written as a sum of two or more consecutive natural numbers

Question

Find the number of ways in which $1000$ can be written as a sum of two or more consecutive natural numbers.

The number of ways of finding the sum of two or more consecutive number will be 1 less than the number of odd factors of 1000. How?

Answer 1

Hint:

For instance, $5$ is a odd number dividing $1000$ since $1000=5*200$.

As a consequence we can find a sum of $5$ consecutive numbers centered on $200$ that sums to $1000$: $1000=198+199+200+201+202$.

• Can you generalize this to any odd number dividing $1000$ ?

• Reversely, can you prove that a list of consecutive numbers summing to $1000$ has an odd number of element ? And that this cardinal divides $1000$ ?

• Finally, the "1 less" comes from the fact that you don't want a single term in the sum, so the the product $1*1000=1000$ doesn't yield a solution.