Question

The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs $(x,y)$ of positive integers is the harmonic mean of $x$ and $y$ equal to $20$

Answer 1

9 ordered pairs

Explanation:

The requirement that the harmonic mean of two positive integers be 20 places a restriction on what those integers may be.

20 = 2/(1/x +1/y)

y = 10x/(x -10)

Integer values of x that produce integer values of y are ...

x ∈ {11, 12, 14, 15, 20, 30, 35, 60, 110}

The corresponding 9 ordered pairs are ...

(11, 110), (12, 60), (14, 35), (15, 30), (20, 20), (30, 15), (35, 14), (60, 12), (110, 11)