AMC 8 2024 Problem 5
AMC8 2024 年第 5 题
基本计数原理★★☆☆☆
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投掷两枚标准的六面殷子。她发现掷出的两个点数之积是6的倍数，请问下列哪个整数不可能是这两个点数之和？
- A.5
- B.6
- C.7
- D.8
- E.9
正确答案 · Correct Answer
B
解析 · Solution
解法1
首先，找出所有乘积为6的数对。然后，通过排除法，我们可以得到以下结果： is possible: 2×3 。 is possible: 1×6 。 is possible: 2×6 Only integer that cannot be the sum is 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
首先，注意到要使两个数的乘积为6，要么其中一个数是6的因子，要么两个数相乘得6（值得注意的是 2×3）。当我们查看选项时，我们看到选项C、D、E都大于6，因此它们都可以相加得到6。选项A成立，因为 2+3=5 ，所以6是唯一不成立的选项。因此，答案是6。
Solution 1
First, figure out all pairs of numbers whose product is 6. Then, using the process of elimination, we can find the following: is possible: 2×3. is possible: 1×6. is possible: 2×6 The only integer that cannot be the sum is 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
First, see that in order for the two numbers to have a product of 6, it must either have 6 as a factor or have factors that have a product of 6 (Notably 2×3 ). When we look at the answer choices, we see that answers C, D, E all are more than 6, so they can all be added as a sum of 6. Answer works because 2+3=5, so 6 is the only answer choice that doesn't work. Thus, the answer is 6.
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