Question

假设

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

是整数。下列关于 $x$ 的哪一项一定为真？

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 $12.$

通分得 $\dfrac{2x}{3} - \dfrac{x}{6} = \dfrac{4x - x}{6} = \dfrac{x}{2}$ 。

要使 $\dfrac{x}{2}$ 为整数， $x$ 必须是偶数。

例如 $x = 4$ ，这说明 $x$ 不必是 $3$ 的倍数，也排除了其他陈述。

所以正确答案是 B。

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.