HMMT February

Note that $A$ is a multiple of $p, q$, and $r$, so $K=\frac{A}{p q r}$ is an integer. Dividing through, we have that $$K=8+\frac{12}{p}=7+\frac{55}{q}=2+\frac{50}{r}$$ Then $p \in\{2,3\}, q \in\{5,11\}$, and $r \in\{2,5\}$. These values give $K \in\{14,12\}, K \in\{18,12\}$, and $K \in$ $\{27,12\}$, giving $K=12$ and $(p, q, r)=(3,11,5)$. We can then compute $A=p q r \cdot K=3 \cdot 11 \cdot 5 \cdot 12=1980$.