AMC12 2000 A

AMC12 2000 A · Q9

AMC12 2000 A · Q9. It mainly tests Averages (mean), Remainders & modular arithmetic.

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered?

Walter 夫人在一个有五名学生的数学班里进行了一次考试。她以随机顺序将分数输入电子表格，每输入一个分数后，表格都会重新计算班级平均分。Walter 夫人注意到，每次输入分数后，平均分总是整数。分数（按从小到大排列）为 $71$、$76$、$80$、$82$ 和 $91$。Walter 夫人最后输入的分数是多少？

(A) 71 71

(B) 76 76

(D) 82 82

(E) 91 91

Answer

Correct choice: (C)

正确答案：（C）

Solution

The first number is divisible by $1$. The sum of the first two numbers is even. The sum of the first three numbers is divisible by $3.$ The sum of the first four numbers is divisible by $4.$ The sum of the first five numbers is $400.$ Since $400$ is divisible by $4,$ the last score must also be divisible by $4.$ Therefore, the last score is either $76$ or $80.$ Case 1: $76$ is the last number entered. Since $400\equiv 76\equiv 1\pmod{3}$, the fourth number must be divisible by $3,$ but none of the scores are divisible by $3.$ Case 2: $80$ is the last number entered. Since $80\equiv 2\pmod{3}$, the fourth number must be $2\pmod{3}$. The only number which satisfies this is $71$. The next number must be $91$ since the sum of the first two numbers is even. So the only arrangement of the scores $76, 82, 91, 71, 80$ or $82, 76, 91, 71, 80$ $\Rightarrow \text{(C)}$

第一个数能被 $1$ 整除。前两个数的和为偶数。前三个数的和能被 $3$ 整除。前四个数的和能被 $4$ 整除。前五个数的和为 $400$。由于 $400$ 能被 $4$ 整除，最后一个分数也必须能被 $4$ 整除。因此最后一个分数只能是 $76$ 或 $80$。情况 1：$76$ 是最后输入的分数。因为 $400\equiv 76\equiv 1\pmod{3}$，第四个数必须能被 $3$ 整除，但这些分数中没有一个能被 $3$ 整除。情况 2：$80$ 是最后输入的分数。因为 $80\equiv 2\pmod{3}$，第四个数必须满足 $2\pmod{3}$。唯一满足这一条件的是 $71$。又因为前两个数的和为偶数，接下来必须输入 $91$。因此分数的唯一排列为 $76, 82, 91, 71, 80$ 或 $82, 76, 91, 71, 80$ $\Rightarrow \text{(C)}$

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