Question

Aaliyah 掷两个标准六面骰子。她注意到掷出的两个数的乘积是六的倍数。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

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 $3$ 。选项中的以下和可以出现：和为 $6$ 且乘积能被 $6$ 整除是不可能的：和为 $6$ 的骰子对是 $(1,5),(2,4),(3,3),(4,2),(5,1)$ ，它们的乘积都不能被 $6$ 整除。 $\begin{gathered} 5=2+3,\quad 7=1+6, \\ 8=2+6,\quad 9=3+6. \end{gathered}$

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 $(1,5),(2,4),(3,3),(4,2),(5,1)$ , and none have product divisible by $6$ .

Thus, B is the correct answer.

2024 AMC 8 第 5 题

答案：B

难度评级：960

视频讲解：

文字解答：

其他年份的第 5 题