Question

假设 $n$ 是三个连续整数的乘积，并且 $n$ 能被 $7$ 整除。下列哪一个不一定是 $n$ 的因数？

Suppose that $n$ is the product of three consecutive integers and that $n$ is divisible by $7.$ Which of the following is not necessarily a divisor of $n?$

$6$

$14$

$21$

$28$

$42$

答案：D

知识点：整除性质因数分解反例

难度评级：1370

解答：

三个连续整数中，一定有一个是 $3$ 的倍数，且乘积 $n$ 也有一个偶数因子，因此能被 $6$ 整除。再加上题设的因数 $7$ ，它一定能被 $6, 14, 21$ 和 $42$ 整除。

但 $28=2^2\cdot7$ 需要两个因数 $2$ ，这不一定保证；例如 $5\cdot6\cdot7=210$ 能被 $7$ 整除，却不能被 $28$ 整除。

所以正确答案是 D。

Among three consecutive integers, at least one is even and one is a multiple of $3,$ so $n$ is divisible by $6.$ With the given factor of $7,$ it is divisible by $6, 14, 21,$ and $42.$

But $28=2^2\cdot7$ requires two factors of $2,$ which is not guaranteed: $5\cdot6\cdot7=210$ is divisible by $7$ but not by $28.$

Thus, the correct answer is D.

2001 AMC 10 第 12 题

答案：D

难度评级：1370

解答：

其他年份的第 12 题