GIVEN: n² is a multiple of 24

24 = (2)(2)(2)(3)

So, n² must have at least three 2's and one 3 in its prime factorization.

What does this tell us about n?

Since n² = (n)(n), we can conclude that n must have at least two 2's and one 3 in its prime factorization.

GIVEN: n² is a multiple of 108

108 = (2)(2)(3)(3)(3)

So, n² must have at least two 2's and three 3's in its prime factorization.

What does this tell us about n?

Since n² = (n)(n), we can conclude that n must have at least one 2 and two 3's in its prime factorization.

So, when we combine both pieces of information, we can see that n must have at least two 2's and two 3's in its prime factorization.

In other words, n = (2)(2)(3)(3)(k), where k is some positive integer

Now let's check the answer choices...

A. 12 = (2)(2)(3)

Since n = (2)(2)(3)(3)(k), we can see that n IS divisible by 12

B. 24 = (2)(2)(2)(3)

Since n = (2)(2)(3)(3)(k), we can see that n need NOT be divisible by 24

C. 36 = (2)(2)(3)(3)

Since n = (2)(2)(3)(3)(k), we can see that n IS divisible by 36

D. 72 = (2)(2)(2)(2)(3)

Since n = (2)(2)(3)(3)(k), we can see that n need NOT be divisible by 72

Answer: A,C

Cheers,

Brent

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Brent Hanneson – Creator of greenlighttestprep.com

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