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

Revision as of 23:31, 8 September 2020

Problem

Let $a$ and $b$ be relatively prime positive integers with $a>b>0$ and $\dfrac{a^3-b^3}{(a-b)^3} = \dfrac{73}{3}.$ 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 relatively prime, $a^3-b^3$ and $(a-b)^3$ are both integers as well. Then, for the given fraction to simplify to $\frac{73}{3}$ , the denominator $(a-b)^3$ must be a multiple of $3.$ Thus, $a-b$ is a multiple of $3$ . Looking at the answer choices, the only multiple of $3$ is $\boxed{\textbf{(C)}\ 3}$ .

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 $70$ .

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

Remarks:

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

Also, we can solve for $a-b$ directly instead of solving for $a$ and $b$ : $a^2-2ab+b^2=149-2(70)=9 \implies a-b=3.$

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.

Solution 3

The first step is the same as above which gives $\frac{a^2+ab+b^2}{a^2-2ab+b^2}=\frac{73}{3}$ .

Then we can subtract $3ab$ and then add $3ab$ to get $\frac{a^2-2ab+b^2+3ab}{a^2-2ab+b^2}=\frac{73}{3}$ , which gives $1+\frac{3ab}{(a-b)^2}=\frac{73}{3}$ . $\frac{3ab}{(a-b)^2}=\frac{70}{3}$ . Cross multiply $9ab=70(a-b)^2$ . Since $a>b$ , take the square root. $a-b=3\sqrt{\frac{ab}{70}}$ . Since $a$ and $b$ are integers and relatively prime, $\sqrt{\frac{ab}{70}}$ is an integer. $ab$ is a multiple of $70$ , so $a-b$ is a multiple of $3$ . Therefore $a=10$ and $b=7$ is a solution. So $a-b=\boxed{\textbf{(C)}\ 3}$

Solution 4

Slightly expanding, we have that $\frac{(a-b)(a^2+ab+b^2)}{(a-b)(a-b)(a-b)}=\frac{73}{3}$ .

Canceling the $(a-b)$ , cross multiplying, and simplifying, we obtain that

$0=70a^2-149ab+70b^2$ . Dividing everything by $b^2$ , we get that

$0=70(\frac{a}{b})^2-149(\frac{a}{b})+70$ .

Applying the quadratic formula....and following the restriction that $a>b>0$ ....

$\frac{a}{b}=\frac{10}{7}$ .

Hence, $7a=10b$ .

Since they are relatively prime, $a=10$ , $b=7$ .

$10-7=\boxed{\textbf{(C)}\ 3}$ .

@@ Line 1: / Line 1: @@
-==Problem 17==
+==Problem==
-Let <math>a</math> and <math>b</math> be relatively prime integers with <math>a>b>0</math> and <math>\frac{a^3-b^3}{(a-b)^3}</math> = <math>\frac{73}{3}</math>. What is <math>a-b</math>?
+Let <math>a</math> and <math>b</math> be relatively prime positive integers with <math>a>b>0</math> and <math>\dfrac{a^3-b^3}{(a-b)^3} = \dfrac{73}{3}.</math> What is <math>a-b?</math>
-<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math>
+<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad \textbf{(E)}\ 5</math>
+==Solution 1==
+Since <math>a</math> and <math>b</math> are relatively prime, <math>a^3-b^3</math> and <math>(a-b)^3</math> are both integers as well. Then, for the given fraction to simplify to <math>\frac{73}{3}</math>, the denominator <math>(a-b)^3</math> must be a multiple of <math>3.</math> Thus, <math>a-b</math> is a multiple of <math>3</math>. Looking at the answer choices, the only multiple of <math>3</math> is <math>\boxed{\textbf{(C)}\ 3}</math>.
+== Solution 2 ==
+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 be seen upon squaring and summing the various factor pairs of <math>70</math>.
+Thus, <math>a - b = \boxed{\textbf{(C)}\ 3}</math>.
+'''Remarks:'''
+An alternate method of solving the system of equations involves solving the second equation for <math>a</math>, by 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>
+Also, we can solve for <math>a-b</math> directly instead of solving for <math>a</math> and <math>b</math>: <math>a^2-2ab+b^2=149-2(70)=9 \implies a-b=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.
+== Solution 3 ==
+The first step is the same as above which gives <math>\frac{a^2+ab+b^2}{a^2-2ab+b^2}=\frac{73}{3}</math>.
+Then we can subtract <math>3ab</math> and then add <math>3ab</math> to get <math>\frac{a^2-2ab+b^2+3ab}{a^2-2ab+b^2}=\frac{73}{3}</math>, which gives <math>1+\frac{3ab}{(a-b)^2}=\frac{73}{3}</math>. <math>\frac{3ab}{(a-b)^2}=\frac{70}{3}</math>.
+Cross multiply <math>9ab=70(a-b)^2</math>. Since <math>a>b</math>, take the square root. <math>a-b=3\sqrt{\frac{ab}{70}}</math>.
+Since <math>a</math> and <math>b</math> are integers and relatively prime, <math>\sqrt{\frac{ab}{70}}</math> is an integer. <math>ab</math> is a multiple of <math>70</math>, so <math>a-b</math> is a multiple of <math>3</math>.
+Therefore <math>a=10</math> and <math>b=7</math> is a solution.
+So <math>a-b=\boxed{\textbf{(C)}\ 3}</math>
+== Solution 4 ==
+Slightly expanding, we have that
+<math>\frac{(a-b)(a^2+ab+b^2)}{(a-b)(a-b)(a-b)}=\frac{73}{3}</math>.
+Canceling the <math>(a-b)</math>, cross multiplying, and simplifying, we obtain that
+<math>0=70a^2-149ab+70b^2</math>.
+Dividing everything by <math>b^2</math>, we get that
+<math>0=70(\frac{a}{b})^2-149(\frac{a}{b})+70</math>.
+Applying the quadratic formula....and following the restriction that <math>a>b>0</math>....
+<math>\frac{a}{b}=\frac{10}{7}</math>.
+Hence, <math>7a=10b</math>.
+Since they are relatively prime, <math>a=10</math>, <math>b=7</math>.
+<math>10-7=\boxed{\textbf{(C)}\ 3}</math>.
+== See Also ==
+{{AMC10 box|year=2012|ab=A|num-b=16|num-a=18}}
+{{MAA Notice}}

2012 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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