Question

How many pairs of positive integers $(a,b)$ are there such that $a$ and $b$ have no common factors greater than $1$ and

$\frac{a}{b}+\frac{14b}{9a}$

is an integer?

$4$

$6$

$9$

$12$

infinitely many

Answer: A

Concepts:divisibility greatest common divisor Diophantine Equation

Difficulty rating: 2170

Solution:

Combining, the expression is $\dfrac{9a^2+14b^2}{9ab}.$ For this to be an integer, $a$ must divide $9a^2+14b^2,$ hence $a\mid 14b^2.$ Since $\gcd(a,b)=1,$ we get $a\mid 14.$ Similarly $b\mid 9a^2$ gives $b\mid 9.$

So $a\in\{1,2,7,14\}$ and $b\in\{1,3,9\}.$ Checking these, only $b=3$ makes the expression an integer for each allowed $a.$

The valid pairs are $(1,3),(2,3),(7,3),$ and $(14,3),$ for a total of $4.$

Thus, the correct answer is A.

2007 AMC 10B Problem 25

Answer: A

Difficulty rating: 2170

Solution:

Problem 25 in Other Years