Question

设 $Z$ 是一个 $6$ 位正整数，例如 $247247$ ，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 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

小提示：

设 $n$ 为重复出现的三位数。用 $n$ 表示 $Z$

Let $n$ be the three-digit block that is repeated. Express $Z$ in terms of $n$ .

大提示：

分解 $1001$

Factor $1001$ .

视频讲解：

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文字解答：

设 $n$ 为任意一个重复数块所组成的三位数。那么 $\begin{aligned} Z&=1000n+n\\ &=1001n\\ &=7\cdot11\cdot13\cdot n \end{aligned}\text{。}$ 因此， $11$ 一定是 $Z$ 的因数。

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