Number Theory Prime Factor Problem

Question

There is an integer N that has 12 factors, including 1 and itself, but only 3 of them are prime factors. The sum of these three prime factors is 20. What is the smallest possible value for N?

Answer 1

If $N$ has prime factors $pqr$ then $N = p^2qr$ since there is only one way to partition $12$ into $3$ parts with each part $> 1$ and the product of these parts begin equal to $12$. Since the sum of the prime factors is even, one of them must be $2$ and to minimize $N, 2 = p$. Then $q = 7, r = 11$ to minimize $N$.