Understand the problem

Given that the number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ is an integer where $p, q$ and $r$ are prime positive numbers, determine a.

Source of the problem

Danube 2014

Topic
Number Theory
Difficulty Level
Easy
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

First settle the case where p,q,r are not all distinct. Also, pqr|pq(p+q)+qr(q+r)+rp(r+p).
From hint 1, it follows that p|q+r, q|p+r, r|p+q.
Hint 2 implies that p,q and r all divide p+q+r. If they are distinct, then pqr|p+q+r. Clearly this is a very strong restriction as it is expected that p+q+r\le pqr.
Show that, if p,q,r are not all equal to 2, then pqr>p+q+r which is a contradiction. Hence the only solution is a=6.

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