Question

Aaliyah 掷两个标准 $6$ 面骰子。她注意到掷出的两个数的乘积是 $6$ 的倍数。下列哪个整数不可能是这两个数的和？

Aaliyah rolls two standard $6$ -sided dice. She notices that the product of the two numbers rolled is a multiple of $6.$ Which of the following integers cannot be the sum of the two numbers?

$5$

$6$

$7$

$8$

$9$

答案：B

知识点：骰子（概率）整除性分类讨论

难度评级：960

小提示：

$6$ 的倍数需要同时有因子 $2$ 和因子 $3$

A multiple of $6$ needs a factor of $2$ and a factor of $3$

大提示：

列出骰子乘积能被 $6$ 整除时可能的和

List the possible sums from dice products divisible by $6$

视频讲解：

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文字解答：

乘积要成为 $6$ 的倍数，两个骰子必须共同提供因子 $2$ 和 $3$ 。选项中的以下各个和都能出现： $\begin{gathered} 5=2+3,\quad 7=1+6, \\ 8=2+6,\quad 9=3+6 \end{gathered}\text{。}$ 但不可能得到和为 $6$ 且乘积能被 $6$ 整除的结果：和为 $6$ 的骰子点数组合是 $(1,5)$ 、 $(2,4)$ 、 $(3,3)$ 、 $(4,2)$ 和 $(5,1)$ ，它们的乘积都不能被 $6$ 整除。

正确答案是 B。

For the product to be a multiple of $6$ , the two dice together must supply a factor of $2$ and a factor of $3$ . The possible sums among the answer choices can occur as follows: $\begin{gathered} 5=2+3,\quad 7=1+6, \\ 8=2+6,\quad 9=3+6. \end{gathered}$ There is no way to get sum $6$ while also having a product divisible by $6$ : the pairs with sum $6$ are $(1,5),$ $(2,4),$ $(3,3),$ $(4,2),$ and $(5,1)$ , and none have product divisible by $6$ .

Thus, B is the correct answer.

2024 AMC 8 第 5 题