2009 AMC 8 第 11 题
先试着解答 2009 AMC 8 第 11 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2009 AMC 8 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
11.
阿马科中学的书店出售铅笔，每支铅笔价格为整数美分。一些七年级学生每人买了一支铅笔，总共支付 美元。 名六年级学生中的一些人每人买了一支铅笔，总共支付 美元。买铅笔的六年级学生比七年级学生多多少人？
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of dollars. Some of the sixth graders each bought a pencil, and they paid a total of dollars. How many more sixth graders than seventh graders bought a pencil?
小提示：
铅笔价格必须同时整除两个以美分表示的总价
The pencil price must divide both total costs in cents.
大提示：
利用最多只有 名六年级学生买铅笔，排除 美分的价格
Use the fact that at most sixth graders bought pencils to rule out a -cent price.
解答：
买铅笔的七年级学生人数等于 除以一支铅笔的价格。同样，买铅笔的六年级学生人数等于 除以一支铅笔的价格。
因此铅笔价格同时整除 和 。质因数分解得 和 同时整除 和 的正整数只有 和 。
如果一支铅笔价格是 美分，那么会有 名六年级学生买铅笔，这是不可能的。因此一支铅笔价格是 美分。
于是 名七年级学生买铅笔， 名六年级学生买铅笔。所以六年级学生比七年级学生多 人。
所以正确答案是 D。
The number of seventh graders that bought a pencil is divided by the price of a pencil. Similarly, the number of sixth graders that bought a pencil is divided by the price of a pencil.
This means that the price of a pencil divides both and Prime factorizing, we get and The only numbers that divide both and are and
If cent was the price of the pencil, that means sixth graders bought pencils, which is not possible. Therefore, the price of a pencil is cents.
This means that seventh graders bought a pencil, and sixth graders bought a pencil. Therefore, more sixth graders than seventh graders bought pencils.
Thus, D is the correct answer.
其他年份的第 11 题
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
- 2002 AMC 8
- 2003 AMC 8
- 2004 AMC 8
- 2005 AMC 8
- 2006 AMC 8
- 2007 AMC 8
- 2008 AMC 8
- 2010 AMC 8
- 2011 AMC 8
- 2012 AMC 8
- 2013 AMC 8
- 2014 AMC 8
- 2015 AMC 8
- 2016 AMC 8
- 2017 AMC 8
- 2018 AMC 8
- 2019 AMC 8
- 2020 AMC 8
- 2022 AMC 8
- 2023 AMC 8
- 2024 AMC 8
- 2025 AMC 8
- 2026 AMC 8