Difference between revisions of "2012 AMC 10A Problems/Problem 17"

(Solution 2)
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Using difference of cubes in the numerator and cancelling out one <math>(a-b)</math> in the numerator and denominator gives <math>\frac{a^2 + ab + b^2}{a^2 - 2ab + b^2} = \frac{73}{3}</math>.
 
Using difference of cubes in the numerator and cancelling out one <math>(a-b)</math> in the numerator and denominator gives <math>\frac{a^2 + ab + b^2}{a^2 - 2ab + b^2} = \frac{73}{3}</math>.
  
Set <math>x = a^2 + b^2</math>, and <math>y = ab</math>.  Then <math>\frac{x + y}{x - 2y} = \frac{73}{3}</math>.  Cross multiplying gives <math>3x + 3y = 73x - 146y</math>, and simplifying gives <math>\frac{x}{y} = \frac{149}{70}</math>.  Since <math>149</math> and <math>70</math> are relatively prime, we let <math>x = 149</math> and <math>y = 70</math>, giving <math>a^2 + b^2 = 149</math> and <math>ab = 70</math>.  Since <math>a>b>0</math>, the only solution is <math>(a,b) = (10, 7)</math>, which can either be seen upon squaring and summing the various factor pairs of <math>70</math>.
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Set <math>x = a^2 + b^2</math>, and <math>y = ab</math>.  Then <math>\frac{x + y}{x - 2y} = \frac{73}{3}</math>.  Cross multiplying gives <math>3x + 3y = 73x - 146y</math>, and simplifying gives <math>\frac{x}{y} = \frac{149}{70}</math>.  Since <math>149</math> and <math>70</math> are relatively prime, we let <math>x = 149</math> and <math>y = 70</math>, giving <math>a^2 + b^2 = 149</math> and <math>ab = 70</math>.  Since <math>a>b>0</math>, the only solution is <math>(a,b) = (10, 7)</math>, which can be seen upon squaring and summing the various factor pairs of <math>70</math>.
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An alternate method of solving the system of equations involves solving the second equation for <math>a</math>, plugging it into the first equation, and solving the resulting quartic equation with a substitution of <math>u = b^2</math>.  The four solutions correspond to <math>(\pm10, \pm7), (\pm7, \pm10)</math>
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Thus, the desired quantity <math>a - b = \boxed{\textbf{(C)}\ 3}</math>.
 
Thus, the desired quantity <math>a - b = \boxed{\textbf{(C)}\ 3}</math>.
  
Note that if you double <math>x</math> and double <math>y</math>, you will get different (but not relatively prime) values for <math>a</math> and <math>b</math> that satisfy the original equation.  
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Note that if you double <math>x</math> and double <math>y</math>, you will get different (but not relatively prime) values for <math>a</math> and <math>b</math> that satisfy the original equation.
  
 
== See Also ==
 
== See Also ==
  
 
{{AMC10 box|year=2012|ab=A|num-b=16|num-a=18}}
 
{{AMC10 box|year=2012|ab=A|num-b=16|num-a=18}}

Revision as of 00:54, 9 February 2012

Problem

Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\frac{a^3-b^3}{(a-b)^3}$ = $\frac{73}{3}$. What is $a-b$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution 1

Since $a$ and $b$ are both integers, so must $a^3-b^3$ and $(a-b)^3$. For this fraction to simplify to $\frac{73}{3}$, the denominator, or $a-b$, must be a multiple of 3. Looking at the answer choices, it is only possible when $a-b=\boxed{\textbf{(C)}\ 3}$.

Solution 2

Using difference of cubes in the numerator and cancelling out one $(a-b)$ in the numerator and denominator gives $\frac{a^2 + ab + b^2}{a^2 - 2ab + b^2} = \frac{73}{3}$.

Set $x = a^2 + b^2$, and $y = ab$. Then $\frac{x + y}{x - 2y} = \frac{73}{3}$. Cross multiplying gives $3x + 3y = 73x - 146y$, and simplifying gives $\frac{x}{y} = \frac{149}{70}$. Since $149$ and $70$ are relatively prime, we let $x = 149$ and $y = 70$, giving $a^2 + b^2 = 149$ and $ab = 70$. Since $a>b>0$, the only solution is $(a,b) = (10, 7)$, which can be seen upon squaring and summing the various factor pairs of $70$.


An alternate method of solving the system of equations involves solving the second equation for $a$, plugging it into the first equation, and solving the resulting quartic equation with a substitution of $u = b^2$. The four solutions correspond to $(\pm10, \pm7), (\pm7, \pm10)$


Thus, the desired quantity $a - b = \boxed{\textbf{(C)}\ 3}$.

Note that if you double $x$ and double $y$, you will get different (but not relatively prime) values for $a$ and $b$ that satisfy the original equation.

See Also

2012 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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All AMC 10 Problems and Solutions
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