NL-NWO 2013 Problem 4

For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$ . For example, $P(20) = 8000$ , as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$ , whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$ . (a) Find all positive integers $n$ satisfying $P(n) = 15n$ . (b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$ .