Difference between revisions of "2008 AMC 10A Problems/Problem 9"

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==Problem==
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#redirect [[2008 AMC 12A Problems/Problem 5]]
Suppose that
 
<cmath>\frac{2x}{3}-\frac{x}{6}</cmath>
 
is an integer. Which of the following statements must be true about <math>x</math>?
 
 
 
<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>
 
 
 
==Solution==
 
<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>
 
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>.
 
 
 
==See also==
 
{{AMC10 box|year=2008|ab=A|num-b=8|num-a=10}}
 

Latest revision as of 22:46, 25 April 2008