8. How many pairs of positive integers $(a, b)$, with $a<b$, are there such that $\operatorname{gcd}(a, b)=2$ and $\operatorname{lcm}(a, b)=660$?
(A) 0
(B) 8
(C) 3
(D) 10
(E) 5
(8) The correct answer is $(B)$.

The product of $a$ and $b$ is $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$. Since $M C D(a, b)=2$, one of the two numbers will have only one factor of 2 in its factorization, while the other will have 2; the other prime factors $3,5,11$ are present in the product $a b$ only once, and will therefore not contribute to the $M C D$; they can be freely distributed on one of the two numbers or the other. The $2^{3}=8$ pairs of integers that are obtained are then:

![](https://cdn.mathpix.com/cropped/2024_04_17_8f8a7950b7c731e81cd0g-2.jpg?height=64&width=1495&top_left_y=1101&top_left_x=269)

Problem proposed by Antonio Fanelli and Riccardo Zanotto.