ISL 1966 Problem 29

A given natural number $N$ is being decomposed in a sum of some consecutive integers.

a.) Find all such decompositions for $N=500.$

b.) How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta$ and $\gamma$ are natural numbers) have? Which of these decompositions contain natural summands only?

c.) Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$

Note by Darij: The $0$ is not considered as a natural number.