14. A number $N$ is the product of three distinct primes. How many distinct factors does $N^{5}$ have?
Solution
216

Let the three distinct prime factors of $N$ be $p, q$ and $r$. Therefore, $N^{5}=p^{5} \times q^{5} \times r^{5}$. Each factor of $N^{5}$ may be written as $p^{a} \times q^{b} \times r^{c}$, where $a, b, c \in\{0,1,2,3,4,5\}$. Since there are 6 choices for the value of each of $a, b, c$ there are $6 \times 6 \times 6=216$ distinct factors of $N^{5}$.