A divisor diversion ....

The number $n$ with exactly 15 divisor is either $n = p^{14}$ or $n = p^2q^4$ , as the number of divisors is $14+1=15)$ or $(2+1)\times (4+1)=15)$ , where $p$ and $q$ are primes But for $n < 10,000$ , $n = p^{14}$ is impossible, as $2^{14}> 10,000$ . The solutions for $n = p^2q^4$ for $n > 10,000$ are as follows:

$\quad \quad 2^23^4 = 324\quad \quad 2^25^4 = 2500\quad \quad 2^27^4 = 9604$

$\quad \quad 3^22^4 = 144 \quad \quad 5^22^4 = 400 \quad \quad 7^22^4 = 784$

$\quad \quad 11^22^4 =1936 \quad \quad 13^22^4 = 2704 \quad \quad 17^22^4 = 4624$

$\quad \quad 19^22^4 = 5776 \quad \quad 23^22^4 = 8464 \quad \quad 3^25^4 = 5625$

$\quad \quad 5^23^4 = 2025 \quad \quad 7^23^4 = 3969 \quad \quad 11^23^4 = 9801$

There are $\boxed{15}$ such positive integers.