# Difference between revisions of "2007 AMC 10B Problems/Problem 25"

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? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }9\qquad\textbf{(D) }12\qquad\textbf{(E) }\text{infinitely many}$

## Solution

### Solution 1

Getting common denominators, we have to find coprime $(a,b)$ such that $9ab|9a^2+14b^2$. Clearly, $3|b$. Since $a$ and $b$ are coprime, $a|9a^2+14b^2 \implies a|14$. Similarly, $b|9$. However, $b$ cannot be $9$ as $81a|81 \cdot 14 + 9a^2$ only has solutions when $3|a$. Therefore, $b=3$ and $a \in \{1,2,7,14\}$. Checking them all, we see that they work and the answer is $\boxed{\mathrm{(A) \ } 4}$.

### Solution 2

Let $x = \frac{a}{b}$. We can then write the given expression as $x+\frac{14}{9x} = k$ where $k$ is an integer. We can rewrite this as a quadratic, $9x^2 - 9kx + 14 = 0$. By the Quadratic Formula, $x = \frac{9k\pm\sqrt{81k^2-504}}{18} = \frac{k}{2}\pm\frac{\sqrt{9k^2-56}}{6}$. We know that $x$ must be rational, so $9k^2-56$ must be a perfect square. Let $9k^2-56 = n^2$. Then, $56 = 9k^2-n^2 = (3k - n)(3k + n)$. The factors pairs of $56$ are $1$ and $56$, $2$ and $28$, $4$ and $14$, and $7$ and $8$. Only $2$ and $28$ and $4$ and $14$ give integer solutions, $k = 5$ and $n = 13$ and $k = 3$ and $n = 5$, respectively. Plugging these back into the original equation, we get $\boxed{\mathrm{(A) \ } 4}$ possibilities for $x$, namely $\frac{1}{3}, \frac{14}{3}, \frac{2}{3},$ and $\frac{7}{3}$.

## See Also

 2007 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 24 Followed byLast question 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions

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