Possibility of finding 3 distinct positive integers whose harmonic mean is an integer

Question

Is it possible to find 3 distinct positive integers a, b, c such that the result abc/(ab + bc + ca) is also an integer?

Answer 1

Yes! Note that the distinct integers $2, 3, 6$ satisfy $2\cdot 3\cdot 6 = 36$ and $$2\cdot 3 + 2\cdot 6 + 3\cdot 6 = 6 + 12 + 18 = 36$$ so the quotient $$\frac{2\cdot 3\cdot 6}{2\cdot 3 + 2\cdot 6 + 3\cdot 6}$$ is equal to $1$.