HMMT February

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.