jmc

First, we find the prime factorization of $180$ to be $2^2\cdot 3^2\cdot 5$. Note that the odd divisors of 180 are precisely the integers of the form $3^a{5}^b$ where $0\le a\le 2$ and $0\le b\le 1$. Note also that distributing $(1+3+9)(1+5)$ yields 6 terms, with each integer of the form $3^a{5}^b$ appearing exactly once. It follows that the sum of the odd divisors of 180 is $(1+3+9)(1+5)=13\cdot 6=\boxed{78}$.