AMC10 2008 A

AMC10 2008 A · Q9

AMC10 2008 A · Q9. 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。以下哪项关于$x$的陈述一定是真的？

(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

Answer (B): Because $\frac{2x}{3}-\frac{x}{6}=\frac{x}{2}$ is an integer, $x$ must be even. The case $x=4$ shows that $x$ is not necessarily a multiple of 3 and that none of the other statements must be true.

答案（B）：因为 $\frac{2x}{3}-\frac{x}{6}=\frac{x}{2}$ 是整数，所以 $x$ 必须是偶数。取 $x=4$ 可知，$x$ 不一定是 3 的倍数，而且其他陈述都不一定为真。

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