A B C and fractions

From the equation $\frac{1}{c} = \frac{1}{a} + \frac{1}{b},$ we get $ab = bc + ac$ . We can re-write this equation as $(a - c)(b - c) = c^2.$ The positive solutions to this equation are of the form $(a,b) = (m + c, n + c)$ , where $mn = c^2$ . Thus, the number of solutions is equal to the number of divisors of $c^2$ .

Thus, we want to find the $c$ so that the number of divisors of $c^2$ is 9. There are two kinds of solutions: $c = pq$ , where $p$ and $q$ are distinct primes, and $c = p^4$ , where $p$ is prime.

The solutions of the form $pq$ are 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 15, 21, 33, 39, 51, 57, 69, 87, 93, 35, 55, 65, 85, 95, 77, and 91. The solutions of the form $p^4$ are 16 and 81. This gives us a total of 32 solutions.