Muhammad's Integers

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

gcd(m3,n2)=2232 and lcm(m2,n3)=243456? \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) \gcd(a, b) and lcm(a,b) \text{lcm}(a, b) denote the greatest common divisor and least common multiple of a a and b b , respectively.

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