Enumerate

First we break the expression into prime factors.

$\begin{aligned} \frac {15^{a+3}28^{8-a}}{35^{2a-9}} & = 3^{a+3}5^{a+3-2a+9}2^{16-2a}7^{8-a-2a+9} = 2^{16-2a}3^{a+3}5^{12-a}7^{17-3a} \end{aligned}$

Since primes do not divide one another, the expression is not an integer when any of the powers of the prime factor is negative. For the powers to be $\ge 0$ , we have:

$\begin{cases} 16-2a \ge 0 & \implies a \le 8 \\ a+3 \ge 0 & \implies a \ge -3 \\ 12-a \ge 0 & \implies a \le 12 \\ 17-3a \ge 0 & \implies a \le 5 \end{cases}$

The acceptable range is $-3 \le a \le 5$ , $\boxed{9}$ integral values.