2008 AMC 12A Problem 5

Combining the fractions, $$\dfrac{2x}{3} - \dfrac{x}{6} = \dfrac{4x - x}{6} = \dfrac{x}{2}.$$ This is an integer exactly when $x$ is even. The example $x = 4$ is even but not a multiple of $3,$ which rules out every other statement. Thus, B is the correct answer.