The number $90$ is a polite number, what is its politeness? 
The number $90$ is a polite number, what is its politeness?
A. $12$
B. $9$
C. $6$
D. $14$
E. $3$

How did you get that answer? I tried Wikipedia to figure out what a polite number was and how to figure out its politeness but I'd like to see it done step by step or have it explained because I just don't understand.

A polite number, it seems, is a positive integer $n$, such that there is a list of consecutive positive integers $a, a+1,\dots, a+r$ with $n = a + (a + 1) + \dots + (a + r)$.
The politeness is the number of representations of a polite number. For example $9$ is polite and its only representations are $2+3+4$ and $4+5$ (as you can verify), so it has politeness $2$. 
The politeness of a number turns out to be the number of its odd divisors, greater than one. For example $9 = 3^2$ has the divisors $1,3,9$, the latter two are odd divisors and greater than one, so again: $9$ has politeness $2$. A prime number $p$ has only $1,p$ as divisors, therefore it has politeness $1$ if and only if it is not $2$ (since $p=2$ is not odd). 
