Question

Walter 老师给一个五人数学班考试。她把成绩按随机顺序输入电子表格，表格在每输入一个成绩后重新计算班级平均分。她注意到每次输入后，平均分总是整数。五个成绩按升序为 $71, 76, 80, 82$ 和 $91$ Walter 老师最后输入的成绩是多少？

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?

$71$

$76$

$80$

$82$

$91$

答案：C

知识点：整除性模运算平均数

难度评级：1560

解答：

$71, 76, 80, 82, 91$ 除以 $3$ 的余数分别是 $2, 1, 2, 1, 1$ 前三个成绩的和必须能被 $3$ 整除，唯一可行的三元组是 $76 + 82 + 91 = 249$ 因此前三个成绩是 $91$ 、 $76$ 、 $82$ 。

因为 $249$ 比 $4$ 的倍数多一，第四个成绩必须比 $4$ 的倍数多三，只有 $71$ 符合；剩下 $80$ 作为第五个成绩。

确实，部分和 $76, 158, 249, 320, 400$ 分别能被 $1, 2, 3, 4, 5$ 整除。

所以正确答案是 C。

The residues of $71, 76, 80, 82, 91$ modulo $3$ are $2, 1, 2, 1, 1.$ The sum of the first three scores must be divisible by $3,$ and the only such triple is $76 + 82 + 91 = 249,$ so the third score entered is $91$ and the first two are $76$ and $82.$

Since $249$ is one more than a multiple of $4,$ the fourth score must be three more than a multiple of $4,$ which only $71$ satisfies. That leaves $80$ as the fifth and last score.

Indeed $76, 158, 249, 320, 400$ are divisible by $1, 2, 3, 4, 5.$

Thus, the correct answer is C.

2000 AMC 10 第 14 题

答案：C

难度评级：1560

解答：

其他年份的第 14 题