6

Part of 2016 Federal Competition For Advanced Students, P2

Problems(1)

An integer sum of three fractions

Source: Austrian Math Olympiad 2016, final round, problem 6

a,b,c

be three integers for which the sum

\frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}

is integer. Prove that each of the three numbers \frac{ab}{c}, \frac{ac}{b}, \frac{bc}{a} is integer.
(Proposed by Gerhard J. Woeginger)

algebranumber theory