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How many factors of 3600 are divisible by 6? A) 45 B) 24 C)

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How many factors of 3600 are divisible by 6? A) 45 B) 24 C)

by GMATinsight » Tue May 29, 2018 6:48 am

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How many factors of 3600 are divisible by 6?

A) 45
B) 24
C) 18
D) 15
E) 9

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by Keith@ThePrincetonReview » Tue May 29, 2018 4:37 pm
GMATinsight wrote:How many factors of 3600 are divisible by 6?

A) 45
B) 24
C) 18
D) 15
E) 9

Source : www.GMATinsight.com
The question asks how many factors of 3,600 are evenly divisible by 6.
In other words, how many factors of 3,600 are multiples of 6.
Listing the factors of 3,600 would probably take too long (there are 43 of them), so work with prime factors instead.
The prime factorization of 3,600 is 2^4 * 3^2 * 5^2.

Imagine a number 'n', which is a factor of 3,600 and also a multiple of 6.
Since 'n' is a multiple of 6, its prime factorization will include at least one 2 and at least one 3.
Remove one 2 and one 3 from the prime factorization of 3,600, leaving 2^3 * 3^1 * 5^2.

If 'n' has no additional prime factors, then n = 2 * 3, which is 6.
However, if 'n' has additional prime factors, those factors must represent some combination of the remaining prime factors of 3,600.
To determine the number of values of 'n' that are possible, calculate the number of combinations that can be made with the remaining prime factors of 3,600.

There are four ways that the number 2 could appear among the remaining prime factors of 'n' (zero 2s, one 2, two 2s, or three 2s).
There are two ways that the number 3 could appear among the remaining prime factors of 'n' (zero 3s or one 3).
There are three ways that the number 5 could appear among the remaining prime factors of 'n' (zero 5s, one 5, or two 5s).
Thus, there are 4 * 2 * 3 = 24 possible combinations of the remaining prime factors of 3,600, and 24 possible values of 'n' (factors of 3,600 that are also multiples of 6).

The correct answer is choice B.
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by Jeff@TargetTestPrep » Thu May 31, 2018 3:24 pm
GMATinsight wrote:How many factors of 3600 are divisible by 6?

A) 45
B) 24
C) 18
D) 15
E) 9
Breaking down 3600 we see that:

3600 = 6 x 600 = 6 x (6 x 100) = 6 x (2 x 3 x 2^2 x 5^2) = 6 x (2^3 x 3^1 x 5^2)

We see that 600 = 2^3 x 3^1 x 5^2, which means 600 has (3+1) x (1+1) x (2+1) = 24 factors. Thus, we can pair 6 with any of the 24 factors of 600 to produce a product that is divisible by 6. Since each of these 24 pairings is a factor of 3,600, we have 24 factors of 3600 that are divisible by 6.

Answer: B

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