Problem 549

Review the given condition as
$$p q (2 r + 50) = p r (7 q + 55) = q r (8 p + 12) = A.$$
This implies that $A$ is a multiple of $p$, $q$ and $r$ so the value $K = \frac{A}{p q r}$ is an integer. Dividing through, we have
$$K = 8 + \frac{12}{p} = 7 + \frac{55}{q} = 2 + \frac{50}{r}.$$
Hence, $p \mid 12$, $q \mid 55$, $r \mid 50$. So we have 3 cases as follow
- $p = 2$, $q = 11$, $r = 5$.
- $p = 3$, $q = 11$, $r = 5$.
- $p = 3$, $q = 5$, $r = 2$.
We can check that only $(p, q, r) = (3, 11, 5)$ works so $K = 12$ and the value of $A$ is $1980$. $\square$