AMC12 2008 A

AMC12 2008 A · Q5

AMC12 2008 A · Q5. It mainly tests Fractions, Parity (odd/even).

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

假设 \[\frac{2x}{3}-\frac{x}{6}\] 是一个整数。关于 $x$，下列哪项陈述一定为真？

(A) It is negative. 它是负数。

(B) It is even, but not necessarily a multiple of 3. 它是偶数，但不一定是3的倍数。

(D) It is a multiple of 6, but not necessarily a multiple of 12. 它是6的倍数，但不一定是12的倍数。

(E) It is a multiple of 12. 它是12的倍数。

Answer

Correct choice: (B)

正确答案：（B）

Solution

\[\frac{2x}{3}-\frac{x}{6}\quad\Longrightarrow\quad\frac{4x}{6}-\frac{x}{6}\quad\Longrightarrow\quad\frac{3x}{6}\quad\Longrightarrow\quad\frac{x}{2}\] For $\frac{x}{2}$ to be an integer, $x$ must be even, but not necessarily divisible by $3$. Thus, the answer is $\mathrm{(B)}$.

\[\frac{2x}{3}-\frac{x}{6}\quad\Longrightarrow\quad\frac{4x}{6}-\frac{x}{6}\quad\Longrightarrow\quad\frac{3x}{6}\quad\Longrightarrow\quad\frac{x}{2}\] 要使 $\frac{x}{2}$ 为整数，$x$ 必须为偶数，但不一定能被 $3$ 整除。因此，答案是 $\mathrm{(B)}$。

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