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$$p$$, $$q$$ are positive integers. The number of ordered pairs $$(p, q)$$, such that the LCM of $$p$$ and $$q$$ is 12 is ______
Correct Answer: 15
$$12 = 2^2 \times 3$$, so $$p = 2^{a_1}3^{b_1}$$ and $$q = 2^{a_2}3^{b_2}$$ with $$\max(a_1, a_2) = 2$$ and $$\max(b_1, b_2) = 1$$. Pairs of exponents with maximum 2 number $$3^2 - 2^2 = 5$$, and with maximum 1 number $$2^2 - 1^2 = 3$$. So there are $$5 \times 3 = 15$$ ordered pairs.
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