How many ways are there of writing XXXX as a sum of positive consecutive integers?

Interesting Yahoo! Answers thread.

I found this discussion very interesting (after coming upon it after a search due to encountering one of these problems on FTW and failing horribly).

Basically, the result is that the number of ways of writing a number as a sum of consecutive integers is the number of odd divisors of that number, minus one. This is an awesome conclusion.

How many ways are there of writing 123123 as a sum of positive consecutive integers?

Therefore, the number of factors (all of which must be odd, because there is no factor of in ) are:

Since there are 32 odd factors of 123123, the number 123123 can be written as a sum of positive consecutive integers in 31 ways.

Each of these ways is one of the odd divisors of 123123 (we subtract 1 because we can’t count the divisor 1 as that would yield only 1 integer, 123123 itself, and we need plural integers anyways).

If we count the even divisors, that will result in some of the divisors being negative. (See thread.)

Ok, lame post, sorry. Guys, do some of the earlier problems, like that staircase one ^_^