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Difference between revisions of "2008 AMC 12A Problems/Problem 5"
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==Problem==
 
==Problem==
 
Suppose that
 
Suppose that
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<cmath>\frac{2x}{3}-\frac{x}{6}</cmath>
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<center>
 
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<math>\frac {2x}{3} - \frac {x}{6}</math>
 
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</center>
 
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is an integer. Which of the following statements must be true about <math>x</math>?
 
is an integer. Which of the following statements must be true about <math>x</math>?
  
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<math>\textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.} \\
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<math>\mathrm{(A)}\ \text{It is negative.}\qquad\mathrm{(B)}\ \text{It is even, but not necessarily a multiple of 3.}\\\qquad\mathrm{(C)}\ \text{It is a multiple of 3, but not necessarily even.}\\\qquad\mathrm{(D)}\ \text{It is a multiple of 6, but not necessarily a multiple of 12.}\\\qquad\mathrm{(E)}\ \text{It is a multiple of 12.}</math>
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\textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.} \\
 
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\textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.} \\
 
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\textbf{(E)}\ \text{It is a multiple of }12\text{.}</math>
 
  
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==Solution==  
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==Solution==
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Since <math>\frac {2x}{3} - \frac {x}{6} = \frac{4x}{6}-\frac{x}{6}=\frac{3x}{6}=\frac{x}{2}</math> is an integer, <math>x</math> must be even <math>\Rightarrow B</math>.  
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<cmath>\frac{2x}{3}-\frac{x}{6}\quad\Longrightarrow\quad\frac{4x}{6}-\frac{x}{6}\quad\Longrightarrow\quad\frac{3x}{6}\quad\Longrightarrow\quad\frac{x}{2}</cmath>
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For <math>\frac{x}{2}</math> to be an integer, <math>x</math> must be even, but not necessarily divisible by <math>3</math>. Thus, the answer is <math>\mathrm{(B)}</math>.
  
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==See Also==  
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==See also==
 
{{AMC12 box|year=2008|num-b=4|num-a=6|ab=A}}
 
{{AMC12 box|year=2008|num-b=4|num-a=6|ab=A}}

Revision as of 23:46, 25 April 2008

Problem

Suppose that \[\frac{2x}{3}-\frac{x}{6}\] is an integer. Which of the following statements must be true about $x$?

$\mathrm{(A)}\ \text{It is negative.}\qquad\mathrm{(B)}\ \text{It is even, but not necessarily a multiple of 3.}\\\qquad\mathrm{(C)}\ \text{It is a multiple of 3, but not necessarily even.}\\\qquad\mathrm{(D)}\ \text{It is a multiple of 6, but not necessarily a multiple of 12.}\\\qquad\mathrm{(E)}\ \text{It is a multiple of 12.}$

Solution

\[\frac{2x}{3}-\frac{x}{6}\quad\Longrightarrow\quad\frac{4x}{6}-\frac{x}{6}\quad\Longrightarrow\quad\frac{3x}{6}\quad\Longrightarrow\quad\frac{x}{2}\] For $\frac{x}{2}$ to be an integer, $x$ must be even, but not necessarily divisible by $3$. Thus, the answer is $\mathrm{(B)}$.

See also

2008 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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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