Problem 1007

Recall that the number $N=p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{k}^{e_{k}}$ (where the $p_{i}$ are distinct primes) has exactly $(e_{1}+1)(e_{2}+1) \cdots(e_{k}+1)$ positive integer divisors including itself. We seek $N<1000$ such that this expression is 30. Since $30=2 \cdot 3 \cdot 5$, we take $e_{1}=1, e_{2}=2, e_{3}=4$. Then we see that $N=5^{1} 3^{2} 4^{2}=720$ is satisfactory.