Question

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

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 $12.$

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

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，所以其他更强的说法不一定成立。

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.