Consecutive Index Help

[Gusto]

I came across the following problem and have not been able to figure out how to solve it. Any help would be very greatly appreciated!!


A whole number can be expressed as a sum of consecutive whole numbers. The number of different ways that a given whole number can be built out of consecutive whole numbers is called its consecutive index.

1. Among the first 10,000 whole numbers, which have the smallest consecutive index? What is that index?
2. Among the first 10,000 whole numbers, which have the largest consecutive index? What is that index?


I know the smallest consecutive index must be 1 (which is trivial), but is there a way to find all numbers with such an index from 0 to 10,000?
And the second question is driving me crazy. I've tried doing some google research but to no avail. Please help alleviate my frustration! Thanks in advance

 

[Ishuda]

I came across the following problem and have not been able to figure out how to solve it. Any help would be very greatly appreciated!!


A whole number can be expressed as a sum of consecutive whole numbers. The number of different ways that a given whole number can be built out of consecutive whole numbers is called its consecutive index.

1. Among the first 10,000 whole numbers, which have the smallest consecutive index? What is that index?
2. Among the first 10,000 whole numbers, which have the largest consecutive index? What is that index?


I know the smallest consecutive index must be 1 (which is trivial), but is there a way to find all numbers with such an index from 0 to 10,000?
And the second question is driving me crazy. I've tried doing some google research but to no avail. Please help alleviate my frustration! Thanks in advance

Let's say the whole numbers are 0, 1, 2, 3, ... [some people start at one so just wanted to make sure we were talking about the same thing]. Starting at zero also makes your statement right at first blush, i.e. 0 and 1 are consecutive whole numbers and add to one.

Suppose j and n-j are whole numbers. They add to n. Since we would want to consider the two number j and n-j the same as n-j and j, we can take j\(\displaystyle \le\)n-j. Given that, what does it mean for j and n-j to be consecutive? What does that imply. Did you change your mind about 1 having the smallest consecutive index?