AMC8 2015

AMC8 2015 · Q22

AMC8 2015 · Q22. It mainly tests Primes & prime factorization, Counting divisors.

On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?

6月1日，一群学生排成每行15人的行。6月2日，同一群学生排成一行长行。6月3日，同一群学生排成每行仅一人。6月4日，同一群学生排成每行6人。此过程持续到6月12日，每天每行学生数不同。然而，6月13日，他们找不到新的组织方式。该群学生的最小可能人数是多少？

(A) 21 21

(B) 30 30

(D) 90 90

(E) 1080 1080

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The number of students must be a multiple of 6 and also a multiple of 15. So the number of students must be divisible by the least common multiple of 6 and 15 which is 30. The divisors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30, so there are only 8 divisors. The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. So 60 has 12 divisors and 60 is the smallest possible number of students.

答案（C）：学生人数必须既是 6 的倍数，也是 15 的倍数。因此学生人数必须能被 6 和 15 的最小公倍数 30 整除。30 的因数有 1、2、3、5、6、10、15、30，共 8 个因数。60 的因数有 1、2、3、4、5、6、10、12、15、20、30、60，共 12 个因数。因此 60 有 12 个因数，并且 60 是可能的最小学生人数。

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