2. 外语
  3. Let S be the set of all positive integers n such that n2 is a multiple of both 24 and 108. Which of the following integers are d

Let S be the set of all positive integers n such that n2 is a multiple of both 24 and 108. Which of the following integers are d

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问题 Let S be the set of all positive integers n such that n2 is a multiple of both 24 and 108. Which of the following integers are divisors of every integer n in S ?
Indicate all such integers.
选项 A、12
B、24
C、36
D、72
答案A，C
解析 To determine which of the integers in the answer choices is a divisor of every positive integer n in S, you must first understand the integers that are in S. Note that in this question you are given information about n2, not about n itself. Therefore, you must use the information about n2 to derive information about n.
    The fact that n2 is a multiple of both 24 and 108 implies that n2 is a multiple of the least common multiple of 24 and 108. To determine the least common multiple of 24 and 108, factor 24 and 108 into prime factors as(23)(3)and(22)(33), respectively. Because these are prime factorizations, you can conclude that the least common multiple of 24 and 108 is(23)(33).
    Knowing that n2 must be a multiple of(23)(33)does not mean that every multiple of(23)(33)is a possible value of n2, because n2 must be the square of an integer. The prime factorization of a square number must contain only even exponents. Thus, the least multiple of(23)(33)that is a square is(24)(34). This is the least possible value of n2, and so the least possible value of n is(22)(32), or 36. Furthermore, since every value of n2 is a multiple of(24)(34), the values of n are the positive multiples of 36; that is, S = {36, 72, 108, 144, 180,...}.
    The question asks for integers that are divisors of every integer n in S, that is, divisors of every positive multiple of 36. Since Choice A, 12, is a divisor of 36, it is also a divisor of every multiple of 36. The same is true for Choice C, 36. Choices B and D, 24 and 72, are not divisors of 36, so they are not divisors of every integer in S. The correct answer consists of Choices A and C.
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