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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
admin
2014-08-12
26
问题
Let S be the set of all positive integers n such that n
2
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 n
2
, not about n itself. Therefore, you must use the information about n
2
to derive information about n.
The fact that n
2
is a multiple of both 24 and 108 implies that n
2
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(2
3
)(3)and(2
2
)(3
3
), respectively. Because these are prime factorizations, you can conclude that the least common multiple of 24 and 108 is(2
3
)(3
3
).
Knowing that n
2
must be a multiple of(2
3
)(3
3
)does not mean that every multiple of(2
3
)(3
3
)is a possible value of n
2
, because n
2
must be the square of an integer. The prime factorization of a square number must contain only even exponents. Thus, the least multiple of(2
3
)(3
3
)that is a square is(2
4
)(3
4
). This is the least possible value of n
2
, and so the least possible value of n is(2
2
)(3
2
), or 36. Furthermore, since every value of n
2
is a multiple of(2
4
)(3
4
), 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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GRE QUANTITATIVE
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