2008 AMC 12A Problems/Problem 5

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 $\Rightarrow B$ .

2008 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2008 AMC 12A Problems/Problem 5

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