Solution:
Let the digits be $a, b, c, d, e$. Then $a b c d e = 180 = 2^{2} \cdot 3^{2} \cdot 5$. We observe that there are 6 ways to factor $180$ into digits $a, b, c, d, e$ (ignoring differences in ordering):
$$180 = 1 \cdot 1 \cdot 4 \cdot 5 \cdot 9 = 1 \cdot 1 \cdot 5 \cdot 6 \cdot 6 = 1 \cdot 2 \cdot 2 \cdot 5 \cdot 9 = 1 \cdot 2 \cdot 3 \cdot 5 \cdot 6 = 1 \cdot 3 \cdot 3 \cdot 4 \cdot 5 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5.$$
There are (respectively) $60, 30, 60, 120, 60$, and $30$ permutations of these breakdowns, for a total of $360$ numbers.