2015 AMC 8 第 22 题
先试着解答 2015 AMC 8 第 22 题，然后核对你的答案与视频与精心整理的解答，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 2015 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。
22.
六月 日，一组学生排成若干行，每行 人。六月 日，同一组学生排成一长行。六月 日，同一组学生每行只站一人。六月 日，同一组学生每行 人。这个过程一直持续到六月 日，每天每行人数都不同。然而到六月 日，他们找不到新的排列方式。这组学生最少可能有多少人？
On June a group of students is standing in rows, with students in each row. On June the same group is standing with all of the students in one long row. On June the same group is standing with just one student in each row. On June the same group is standing with students in each row. This process continues through June with a different number of students per row each day. However, on June they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
小提示：
不同的每行人数就是总人数的因数
The different row sizes are the divisors of the group size.
大提示：
需要最小的同时为 和 的倍数，并且恰好有 个因数
You need the smallest multiple of both and with exactly divisors.
视频讲解：
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文字解答：
每行可站的人数正好是学生总数的正因数。因为六月 日至六月 日有不同排列，而六月 日没有新的排列方式，所以总人数必须恰好有 个正因数。
总人数必须同时能被 和 整除，因此能被 整除。而这个数只有 个因数。
最小的 的倍数且有 个因数的是 ，其因数个数为 。
所以正确答案是 C。
The possible numbers of students per row are exactly the positive divisors of the total number of students. Since June through June give different arrangements and June gives no new one, the total number of students must have exactly positive divisors.
The number must be divisible by both and hence by This number has only divisors.
The smallest multiple of with divisors is which has divisors.
Thus, C is the correct answer.
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