jmc

There are six possible square powers of two that can divide the given number: $2^0$, $2^2$, $2^4$, $2^6$, $2^8$, and $2^{10}$. Similarly, there are seven possible square powers of three that can divide the given number, and eight possible square powers of five that can divide the given number. Therefore, there are $6\times 7\times 8=\boxed{336}$ positive perfect square integers that can divide $\left(2^{10}\right)\left(3^{12}\right)\left(5^{15}\right)$.