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greatest common factor

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by himu » Tue Feb 26, 2013 9:15 pm
The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?

3
14
30
42
70
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by Anurag@Gurome » Tue Feb 26, 2013 9:39 pm
himu wrote:The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
GCF of 16 and n is 4 ---> n is multiple of 2 and 4 but not of 8 or 16
GCF of 45 and n is 3 ---> n is multiple of 2 but not of 5 or 9

Hence, n = 2*2*3*(Something)
And, 210 = 2*3*5*7

Hence, we know for sure that GCF of 210 and n will be a multiple of both 2 and 3 but not of 4 or 5. But it could be a multiple of 7.

Hence, possible GCF of n and 210 are 2*3 = 6, 2*3*7 = 42 etc

The correct answer is D.
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by GMATGuruNY » Wed Feb 27, 2013 6:46 am
himu wrote:The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?

3
14
30
42
70
210 = 2*3*5*7.
We need to determine which of these prime factors could be shared by n.
Since the GCF of n and 16 is 4, n is a multiple of 2.
Since the GCF of n and 45 is only 3, n a multiple of 3 but NOT of 5.
There is no condition that prevents n from being a multiple of 7.
Thus, of the prime factors of 210, n could share all but 5:
2*3*7 = 42.

The correct answer is D.
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by ceilidh.erickson » Wed Feb 27, 2013 8:57 am
To add to what my fellow experts have said...

When you see greatest common factor (GCF) language, it's telling you not only about what these numbers contain, but what they do not contain. Try to put it in this context:

GCF of 16 and n is 4. n contains 2*2. n does not contain more than two 2's.

GCF of n and 45 is 3. n contains one 3. n does not contain a 5 or more than one 3.

You can create a Venn diagram to see the overlap between 210 and n:

Image

We know that n and 210 definitely share a 2 and 3 (eliminate answers A and B), and they definitely don't share a 5 (eliminate C and E). They could share a 7, though, so the correct answer is D.
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