2008 AMC 10A Problem 9

Combining over a common denominator, $\dfrac{2x}{3} - \dfrac{x}{6} = \dfrac{4x - x}{6} = \dfrac{x}{2}.$ For $\dfrac{x}{2}$ to be an integer, $x$ must be even. The example $x = 4$ shows that $x$ need not be a multiple of $3$ and rules out the other statements. Thus, the correct answer is B.