AMC8 2024

AMC8 2024 · Q5

AMC8 2024 · Q5. It mainly tests Probability (basic), Divisibility & factors.

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?

Aaliyah 掷两个标准的六面骰子。她注意到两个骰子数字的乘积是 6 的倍数。下列哪个整数不可能是两个数字的和？

(A) 5 5

(B) 6 6

(D) 8 8

(E) 9 9

Answer

Correct choice: (B)

正确答案：（B）

Solution

First, figure out all pairs of numbers whose product is 6. Then, using the process of elimination, we can find the following: $\textbf{(A)}$ is possible: $2\times 3$ $\textbf{(C)}$ is possible: $1\times 6$ $\textbf{(D)}$ is possible: $2\times 6$ $\textbf{(E)}$ is possible: $3\times 9$ The only integer that cannot be the sum is $\boxed{\textbf{(B) } 6}$. We can prove that B is correct because the only ordered pairs that add to 6 is $(1,5)$ , $(5,1)$, $(2,4)$, $(4,2)$, and ,$(3,3)$, and these multiply to 5, 8, and 9. Slight changes ~ VersatileGopher38

首先，找出所有乘积是 6 倍数的数字对。然后，通过排除法，我们可以得到： $\textbf{(A)}$ 可能：$2\times 3$ $\textbf{(C)}$ 可能：$1\times 6$ $\textbf{(D)}$ 可能：$2\times 6$ $\textbf{(E)}$ is possible: $3\times 9$ 唯一不可能的整数是 $\boxed{\textbf{(B) } 6}$。我们可以证明 B 是正确的，因为和为 6 的有序对只有 $(1,5)$、$(5,1)$、$(2,4)$、$(4,2)$ 和 $(3,3)$，它们的乘积分别是 5、8 和 9。轻微修改 ~ VersatileGopher38

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