How many ordered pairs of positive integers $$(m, n)$$ satisfy

$\gcd (m^3, n^2) = 2^2 \cdot 3^2 \text{ and } \text{lcm}(m^2, n^3) = 2^4 \cdot 3^4 \cdot 5^6?$

This problem is shared by Muhammad A.

Details and assumptions

$$\gcd(a, b)$$ and $$\text{lcm}(a, b)$$ denote the greatest common divisor and least common multiple of $$a$$ and $$b$$, respectively.

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