Solution:
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 $\operatorname{GCD}(60,42)=6$. 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 $\operatorname{GCD}(A, B)=6$, 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 $\operatorname{GCD}(A, C)=3$. Thus, $C$ is a multiple of $21$, and we cannot have $C=42$ or else $\operatorname{GCD}(A, C)$ would be $6$. Thus, $C=21$, giving the solution $A=6$, $B=20$, $C=21$.