Thrice the Product

$\begin{aligned} 3AB & = \overline{AB} \\ \implies 3AB & = 10A+B \\ 3AB - 10A & = B \\ \implies A & = \frac B{3B-10} \end{aligned}$

We note that for $A$ to be a positive integer, $B \ge 4$ . For $B=4$ , $\implies A = 2$ and $B=5$ , $\implies A = 1$ . For $B \ge 6$ , $\dfrac B{3B-10} < 1$ . Therefore, there are only $\boxed{2}$ two-digit positive integers 15 and 24 that satisfy the equation above.