Question

设 $Z$ 是一个六位正整数，例如二十四万七千二百四十七，其前三位数字与后三位数字按相同顺序完全相同。下列哪个数一定也是 $Z$ 的因数？

Let $Z$ be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z?$

$11$

$19$

$101$

$111$

$1111$

答案：A

知识点：质因数分解位值

难度评级：940

视频讲解：

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文字解答：

设 $Z$ 的形式为 $n$ 。则这说明 $11$ 一定是的因数。 $\begin{aligned} Z&=1000n+n\\ &=1001n\\ &=7\cdot11\cdot13\cdot n. \end{aligned}$

所以正确答案是 A。

Let $n$ be the three-digit number formed by either repeated block. Then $\begin{aligned} Z&=1000n+n\\ &=1001n\\ &=7\cdot11\cdot13\cdot n. \end{aligned}$ Therefore, $11$ must be a factor of $Z.$

Thus, A is the correct answer.

2017 AMC 8 第 7 题

答案：A

难度评级：940

视频讲解：

文字解答：

其他年份的第 7 题