Library / /4 of 10

, 2018

Number theory Difficulty 5.0 AIME Prove it Saudi Arabia

Distinct prime numbers $p$ , $q$ , $r$ satisfy the equation
$2 p q r + 50 p q = 7 p q r + 55 p r = 8 p q r + 12 q r = A$
for some positive integer $A$ . Find $A$ .

Solution

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 $1980$ . $\square$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.