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# The greatest common factor of 16 and the positive integer n

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The greatest common factor of 16 and the positive integer n [#permalink]  25 Feb 2011, 07:21
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35% (medium)

Question Stats:

52% (02:12) correct 47% (00:58) wrong based on 55 sessions
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?

A. 3
B. 14
C. 30
D. 42
E. 70
[Reveal] Spoiler: OA
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Re: GCF [#permalink]  25 Feb 2011, 07:32
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Expert's post
rosgmat 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?

a) 3
b) 14
c) 30
d) 42
e) 70

The greatest common factor of 2^4=16 and n is 4 --> n is a multiple of 2^2=4 but not the higher powers of 2, for example 2^3=8 or 2^4=16, because if it were then the greatest common factor of 16 and n would be more than 4;

The greatest common factor of 3^2*5=45 and n is 3 --> n is a multiple of 3 but not the higher powers of 3 and not 5, because if it were then the greatest common factor of 3^2*5=45 and n would be more than 3;

So, n is a multiple of 2^2*3=12, not a multiple of higher powers of 2 or 3, and not a multiple of 5 (so n=12x where x could be any positive integer but 2, 3, or 5). Now, as 210=2*3*5*7 then the greatest common factor of n and 210 could be 6 or 6*7=42 (if 7 is a factor of n).

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Re: GCF [#permalink]  25 Feb 2011, 08:34
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You can do the prime boxes.

Prime box of 16: 2, 2, 2, 2
Prime box of 45: 3, 3, 5

Prime box of 210: 2, 5, 3, 7

So, n has at least two 2's and one 3, but n hasn't got any 5. Now, checking alternatives:
A) wrong, as n and 210 share at least one 2 and one 3.
B) wrong again, no 3 in 14.
C) wrong, as 30 has a 5
D) correct. 42 prime box is 2, 3, 7, so it meets all requirements.
E) wrong, 70 prime box has 2, 7 and 5
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Question on GCF [#permalink]  04 Nov 2012, 05:43
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?
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Re: Question on GCF [#permalink]  04 Nov 2012, 05:49
the Common GCF of 16 and n being 4, made me choose n to be 12.
the Common GCF of n and 45 being 3, n= 12 seems to be a valid option here as well.

Hence, the Common GCF of n and 210, i.e. 12 and 210 seems to be 6.
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Re: Question on GCF [#permalink]  04 Nov 2012, 07:12
Some2609 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?

Haha, I just clicked wrong in the poll, but imho here goes the correct way:

16 and n - GCF = 4 = 2 x 2
45 and n - GCF = 3

210 = 2 x 3 x 5 x 7

Eliminate prime factors that are not included in the given options and approve the ones that appear.
Eliminate: 5
Approve: 2, 3, 7

2 x 3 x 7 = 42
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Re: Question on GCF [#permalink]  04 Nov 2012, 07:40
SirGMAT wrote:
Some2609 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?

Haha, I just clicked wrong in the poll, but imho here goes the correct way:

16 and n - GCF = 4 = 2 x 2
45 and n - GCF = 3

210 = 2 x 3 x 5 x 7

Eliminate prime factors that are not included in the given options and approve the ones that appear.
Eliminate: 5
Approve: 2, 3, 7

2 x 3 x 7 = 42

Got it!!...Hey, Thanks a lot!
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Re: Question on GCF [#permalink]  04 Nov 2012, 14:01
Expert's post
Some2609 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?

Merging similar topics. Refer to the solutions above and ask if anything remains unclear.

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Re: Question on GCF   [#permalink] 04 Nov 2012, 14:01
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