Question

Suppose that

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

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

It is negative.

It is even, but not necessarily a multiple of $3.$

It is a multiple of $3,$ but not necessarily even.

It is a multiple of $6,$ but not necessarily a multiple of $12.$

It is a multiple of $12.$

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.