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2007 AMC 12B Problems/Problem 24

Revision as of 13:12, 29 November 2014 by Abvenkgoo (talk | contribs) (→‎Solution 3)

Problem 24

How many pairs of positive integers $(a,b)$ are there such that $\gcd(a,b)=1$ and \[\frac{a}{b}+\frac{14b}{9a}\] is an integer?

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

Solution

Combining the fraction, $\frac{9a^2 + 14b^2}{9ab}$ must be an integer.

Since the denominator contains a factor of $9$, $9 | 9a^2 + 14b^2 \quad\Longrightarrow\quad 9 | b^2 \quad\Longrightarrow\quad 3 | b$

Since $b = 3n$ for some positive integer $n$, we can rewrite the fraction(divide by $9$ on both top and bottom) as $\frac{a^2 + 14n^2}{3an}$

Since the denominator now contains a factor of $n$, we get $n | a^2 + 14n^2 \quad\Longrightarrow\quad n | a^2$.

But since $1=gcd(a,b)=gcd(a,3n)=gcd(a,n)$, we must have $n=1$, and thus $b=3$.

For $b=3$ the original fraction simplifies to $\frac{a^2 + 14}{3a}$.

For that to be an integer, $a$ must divide $14$, and therefore we must have $a\in\{1,2,7,14\}$. Each of these values does indeed yield an integer.

Thus there are four solutions: $(1,3)$, $(2,3)$, $(7,3)$, $(14,3)$ and the answer is $\mathrm {(A)}$

Solution 2

Let's assume that $\frac{a}{b} + \frac{14b}{9a} = m}$ (Error compiling LaTeX. Unknown error_msg) We get--

$9a^2 - 9mab + 14b^2 = 0$

Factoring this, we get $4$ equations-

$(3a-2b)(3a-7b) = 0$

$(3a-b)(3a-14b) = 0$

$(a-2b)(9a-7b) = 0$

$(a-b)(9a-14b) = 0$

(It's all negative, because if we had positive signs, $a$ would be the opposite sign of $b$)

Now we look at these, and see that-

$3a=2b$

$3a=b$

$3a=7b$

$3a=14b$

$a=2b$

$9a=7b$

$a=b$

$9a=14b$

This gives us $8$ solutions, but we note that the middle term needs to give you back $9m$.

For example, in the case

$(a-2b)(9a-7b)$, the middle term is $-25ab$, which is not equal by $-9m$ for whatever integar $m$.

Similar reason for the fourth equation. This elimnates the last four solutions out of the above eight listed, giving us 4 solutions total $\mathrm {(A)}$

Solution 3

Let $u = \frac{a}{b}$. Then the given equation becomes $u + \frac{14}{9u} = \frac{9u^2 + 14}{9u}$.

Let's set this equal to some value, $a \Rightarrow \frac{9u^2 + 14}{9u} = a$.

Clearing the denominator and simplifying, we get a quadratic in terms of $u$:

$9u^2 - 9au + 14 = 0 \Rightarrow u = \frac{9a \pm \sqrt{(9a)^2 - 4\cdot 9 \cdot 14}}{18}$

See Also

2007 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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 12 Problems and Solutions

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