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

(New page: ==Problem== Suppose that <math>\frac {2x}{3} - \frac {x}{6}</math> is an integer. Which of the following statements must be true about <math>x</math>? <math>\textbf{(A)}\ \text{It is ne...)
 
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Suppose that
 
Suppose that
  
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<math>\frac {2x}{3} - \frac {x}{6}</math>
 
<math>\frac {2x}{3} - \frac {x}{6}</math>
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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>?

Revision as of 20:15, 18 February 2008

Problem

Suppose that

$\frac {2x}{3} - \frac {x}{6}$

is an integer. Which of the following statements must be true about $x$?

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

Solution

Since $\frac {2x}{3} - \frac {x}{6} = \frac{4x}{6}-\frac{x}{6}=\frac{3x}{6}=\frac{x}{2}$ is an integer, $x$ must be even $\Rightarrow B$.

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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