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Let

r

and

s

be positive integers. Derive a formula for the number of ordered quadruples

(a,b,c,d)

of positive integers such that

3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),

depending only on

r

and

s.

Problem Statement