Muhammad's Integers

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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