AMC 8 2015 Problem 22
AMC8 2015 年第 22 题
整除与因数★★★☆☆
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?
7 月 1 号那天，一群学生站成几行，每行 15 个学生；7 月 2 号那天，这群学生站成一行；7 月 3 号那天，这群学生站成若干行，每行只有 1 个学生；7 月 4 号那天，这群学生站成若干行，每行 6 个学生。这个过程一直持续到 7 月 12 号，每天每行的人数都不一样。然而，到了 7 月 13 号这天，他们再也没有办法找到一种新的方式去安排这些学生，那么这群学生最少有多少人？
- A.21
- B.30
- C.60
- D.90
- E.1080
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
Since we know the number must be a multiple of 15, we can eliminate . We also know that after 12 days, the students can't find any more arrangements, meaning the number has 12 factors. Now, we just list the factors of every number, starting with 30: 60 has 12 factors, so the answer is 60.
We know that the number of students in the group has to be a multiple of 15, so we can eliminate A. However, on June 13, they cannot find a new way of organizing the students meaning that the number has only 12 factors. The prime factorization of 60 is equal to Using the factor counting formula, we have the add one to each exponent and multiply them together to find the number of factors. factors making 60 the answer.
~ CHECKMATE2021 On the 1st day, all the students are in an arrangement such that there are 15 students per row. On the 4th day, all the students are in an arrangement such that there are 6 students per row. This tells us that the # of students is a multiple of both 15 and 6. Since we can arrange the students for 12 days, this tells us that the # of students has 12 factors. We can also tell that the # of students is a multiple of 30, as 30 is the $LCM$ of 6 and 15. We now test answer choice B. 30 = $2*3*5$ => 30 has $2*2*2 = 8$ factors. Let's test answer choice C. 60 = $2^2*3*5$ => 60 has $3*2*2 = 12$ factors. We know that 60 is our answer. ~ TealBear16🐰
继续练习