Difference between revisions of "2008 AMC 12A Problems/Problem 5"

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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>?
  
<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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<math>\mathrm{(A)}\ \text{It is negative.}\\\mathrm{(B)}\ \text{It is even, but not necessarily a multiple of 3.}\\mathrm{(C)}\ \text{It is a multiple of 3, but not necessarily even.}\\mathrm{(D)}\ \text{It is a multiple of 6, but not necessarily a multiple of 12.}\\mathrm{(E)}\ \text{It is a multiple of 12.}</math>
  
 
==Solution==
 
==Solution==
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{{AMC12 box|year=2008|ab=A|num-b=4|num-a=6}}
 
{{AMC12 box|year=2008|ab=A|num-b=4|num-a=6}}
 
{{AMC10 box|year=2008|ab=A|num-b=8|num-a=10}}
 
{{AMC10 box|year=2008|ab=A|num-b=8|num-a=10}}
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{{MAA Notice}}

Latest revision as of 02:17, 5 August 2021

The following problem is from both the 2008 AMC 12A #5 and 2008 AMC 10A #9, so both problems redirect to this page.

Problem

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

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

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 (ProblemsAnswer KeyResources)
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
2008 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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 10 Problems and Solutions

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