题目信息
In an auditorium, 360 chairs are to be set up in a rectangular arrangement with x rows of exactly y chairs each. If the only other restriction is that 10 < x < 25, how many different rectangular arrangements are possible?
A:Four
B:Five
C:Six
D:Eight
E:Nine
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正确答案：
B:Five
Arithmetic Properties of integers
Because a total of 360 chairs are distributed in x rows of exactly y chairs each, it follows that xy = 360. Also, 10 < x < 25, and so x can only be an integer factor of 360 = 23 × 32 × 5 that is between 10 and 25. Below is a list of all integers from 11 through 24. Since 2, 3, and 5 are the only prime factors of 360, any integer having a prime factor other than 2, 3, or 5 cannot be a factor of 360 and has been crossed out. For example, 21 = 3 × 7 has a prime factor of 7, and thus 21 has been crossed out.
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24
Of the six integers remaining, it is clear that each is a factor of 360 = 23 × 32 × 5 except for 16 = 24. Therefore, the number of possible rectangular arrangements is five.
The correct answer is B.
Because a total of 360 chairs are distributed in x rows of exactly y chairs each, it follows that xy = 360. Also, 10 < x < 25, and so x can only be an integer factor of 360 = 23 × 32 × 5 that is between 10 and 25. Below is a list of all integers from 11 through 24. Since 2, 3, and 5 are the only prime factors of 360, any integer having a prime factor other than 2, 3, or 5 cannot be a factor of 360 and has been crossed out. For example, 21 = 3 × 7 has a prime factor of 7, and thus 21 has been crossed out.
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24
Of the six integers remaining, it is clear that each is a factor of 360 = 23 × 32 × 5 except for 16 = 24. Therefore, the number of possible rectangular arrangements is five.
The correct answer is B.
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