TITLE: Positive rationals satisfying $a+b+c+d=abcd$?
QUESTION [4 upvotes]: Victor has posted a couple of problems involving finding real and rational solutions of $a+b+c=abc$. Two techniques have been given: using triangles, and using scaling. Neither seems to work for the following problem.
How can one easily (without brute force) characterize and produce the quadruples of positive rational numbers such that $a+b+c+d=abcd$?
The triangle technique doesn't seem to work for $n=4$. The scaling technique works, but doesn't necessarily give rational numbers.

REPLY [4 votes]: Write the equation as $a(bcd-1)=b+c+d$.  For any positive rationals $b,c,d$ with $bcd>1$, $a=(b+c+d)/(bcd-1)$ is a positive rational and this gives you a solution.  On the other hand, there is no positive rational solution with $bcd\le 1$.
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