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Let S be the set of all positive integers n such that n^2 is a multipl

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Re: Let S be the set of all positive integers n such that n^2 is a multipl [#permalink]
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Bunuel wrote:
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 ?

A. 12
B. 24
C. 48
D. 72
E. 120

Solution:

If n^2 is a multiple of both 24 and 108, then n^2 is a multiple of the LCM of 24 and 108, which is 216.
Now, let’s prime factorize 216:

216 = 24 x 9 = 2^3 x 3 x 3^2 = 2^3 x 3^3

Since n^2 must have an even number of each of its prime factors, the smallest value of n^2 is 2^4 x 3^4 and thus the smallest value of n is 2^2 x 3^2 = 4 x 9 = 36. We see that of all the given answer choices, only 12 is a divisor of n = 36; thus, 12 is the correct answer.

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Re: Let S be the set of all positive integers n such that n^2 is a multipl [#permalink]
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Re: Let S be the set of all positive integers n such that n^2 is a multipl [#permalink]
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