Question

How many pairs of positive integers (a, b) are there such that a and b have no common factor greater than 1 and $\frac{a}{b}+\frac{14b}{9a}$ is an integer?

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is D

4

Let $u=\frac{a}{b}.$ Then the problem is equivalent to finding all possible rational numbers 'u' such that
$u+\frac{14}{9u}=k$, for some integer k.

This equation is equivalent to $9u^2-9uk+14=0$
$9u^2-9uk+14=0$ whose solutions are,

$u=\frac{9k\pm \sqrt{81k^2-504}}{18}$
$=\frac{k}{2}\pm \frac{1}{6}\sqrt{9k^2-56}$

Hence u is rational if and only if:
$\sqrt{9k^2-56}$ is rational, which is true if and only if $9k^2-56$ is a perfect square.
Suppose that $9k^2-56=S^2$ for some positive integer S.
(3k+S)(3k-S)=56
The only factors of 56 are 1,2,4,7,8,14,28 and 56.
So, (3k-S) and (3k+S) is one of the ordered pairs (1,56), (2,28), (4,14) and (7,8). The cases (1,56) and (7,8) yield no integral solutions.
The cases (2, 28) and (4, 14) yield k = 5 and k = 3 respectively.

If k=5, then $u=\frac{1}{3}$ or $u=\frac{14}{3}.$
If k=3, then $u=\frac{2}{3}$ or $u=\frac{7}{3}.$
Therefore there are four pairs (a,b) that satisfy the given conditions, namely (1,3), (2,3), (7,3) and (14,3).