Question

假设 $\dfrac{2x}{3} - \dfrac{x}{6}$ 是整数。下列关于 $x$ 的说法哪一个一定正确？

Suppose that $\dfrac{2x}{3} - \dfrac{x}{6}$ is an integer. Which of the following statements must be true about $x?$

它是负数。

It is negative.

它是偶数，但不一定是 $3$ 的倍数。

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

它是 $3$ 的倍数，但不一定是偶数。

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

它是 $6$ 的倍数，但不一定是 $12$ 的倍数。

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

它是 $12$ 的倍数。

It is a multiple of $12.$

把两个分数通分合并。

Combine the two fractions over a common denominator

化简为 $\dfrac{x}{2}$ ，再判断它在什么整除条件下是整数。

Simplify to $\dfrac{x}{2},$ then decide what divisibility condition makes this an integer

通分合并可得 $\dfrac{2x}{3} - \dfrac{x}{6} = \dfrac{4x - x}{6} = \dfrac{x}{2}\text{。}$

该式为整数当且仅当 $x$ 是偶数。例如 $x = 4$ 是偶数但不是 $3$ 的倍数，这就排除了其余每一个说法。

所以正确答案是 B。

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.