Let gcd mean the greatest common divisor of two numbers and lcm their least common multiple. Suppose the three numbers $A, B, C$ satisfy

$$
\begin{array}{ll}
\operatorname{gcd}(A, B)=2, & \operatorname{lcm}(A, B)=60 \\
\operatorname{gcd}(A, C)=3, & \operatorname{lcm}(A, C)=42
\end{array}
$$

Determine the three numbers.
From the given information, $A$ must be a multiple of 2 and 3 , and thus a multiple of $\operatorname{LCM}(2,3)=6$. It also must be a factor of 60 and 42 , and thus a factor of $G C D(60,42)=6$. The only possibility is $A=6$.

Since $\operatorname{LCM}(A, B)$ is divisible by 5 but $A$ is not, $B$ must be divisible by 5 . Similarly, since $\operatorname{LCM}(A, B)$ is divisible by $2^{2}=4$ but $A$ is not, $B$ must also be a multiple of 4 and thus a multiple of 20 . $B$ cannot be 60 or we would have $G C D(A, B)=6$, thus $B=20$.

Finally, since $\operatorname{LCM}(A, C)$ is a multiple of 7 but $A$ is not, $C$ must be divisible by 7 . Also, $C$ is divisible by 3 since $G C D(A, C)=3$. Thus, $C$ is a multiple of 21 , and we cannot have $C=42$ or else $G C D(A, C)$ would be 6 . Thus, $C=21$, giving the solution $A=6, B=20, C=21$.