2024 AMC 8 第 5 题

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.