AMC 8 2009 Problem 11
AMC8 2009 年第 11 题
整除与因数★★★☆☆
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43. Some of the 30 sixth graders each bought a pencil, and they paid a total of $1.95. How many more sixth graders than seventh graders bought a pencil?
Amaco 中学的书店所卖的铅笔每根的价格是个整数（单位：美分）。一些七年级学生每个人买了一支铅笔，总共花了 1.43 美元。一些六年级学生（不超过 30 人）每人买了一支铅笔，总共花了 1.95 美元。买铅笔的人当中，六年级学生比七年级多多少人？
- A.1
- B.2
- C.3
- D.4
- E.5
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
The number of sixth graders who bought a pencil is 195 195 divided by the cost of a pencil. Similarly the number of seventh graders who bought a pencil is 143 143 divided by the cost of a pencil. That means both 195 195 and 143 143 are multiples of the price of the pencil. Factor 195 = 1 ⋅ 3 ⋅ 5 ⋅ 13 195=1⋅3⋅5⋅13 and 143 = 1 ⋅ 11 ⋅ 13 143=1⋅11⋅13. The only common divisors are 1 1 and 13 13. If a pencil cost 1 1 cent, then 195 195 sixth graders bought a pencil. However, there are only 30 30 sixth graders, so a pencil must cost 13 13 cents. Using that fact, 195 13 − 143 13 = 15 − 11 = 4 13 195 − 13 143 =15−11=4 more sixth graders than seventh graders bought pencils. Answer: 𝐷 D .
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