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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 $a$.

Danube 2014

Number Theory
Easy
Suggested Book
An Excursion in Mathematics

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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