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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]  22 Feb 2011, 17:01
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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: The greatest common factor of 16 and the positive integer n [#permalink]  22 Feb 2011, 18:36
EDIT.... My explanation was wrong :p [Correcting it]

Corrected Version:

Let's try the Prime Box Approach

Prime Box is simply a collection of all prime factors of a given number!

(1) Prime Box of 16 = |2, 2, 2, 2|
(2) Prime Box of 45 = |3, 3, 5|

(3) Prime Box of n = |2, 2, 3....|
(4) Prime Box of 210 = |2, 5, 3, 7|

From 3 and 4:
The GCF of n and 210 must be a multiple of 6.

So we can eliminate A, B and E!

From 2 and 3:
n is not a multiple of 5. If it were, the GCF of n and 45 would have been 15!
So we can eliminate C

The only remaining choice is 'D'
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Last edited by AmrithS on 22 Feb 2011, 18:51, edited 2 times in total.
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Re: The greatest common factor of 16 and the positive integer n [#permalink]  22 Feb 2011, 18:38
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Expert's post
ajit257 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

i am not so sure about the oa.

GCF (n, 16) = 4
This means 4 is a factor of n but 8 and 16 are not. (If 8 were a factor of n too, the GCF would have been 8. Similarly for 16)

GCF (n, 45) = 3
This means 3 is a factor of n but 9 and 5 are not. Same logic as above.

210 = 2*3*5*7
n has 4 and 3 as factors and it doesn't have 5 as a factor.
so GCF of n and 210 could be 6 (if 7 is not a factor of n) or 42 (if 7 is a factor of n)

Note: 3 is definitely not the GCF of n and 210 because they definitely have 3*2 in common. So GCF has to be at least 6.
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Re: The greatest common factor of 16 and the positive integer n [#permalink]  22 Feb 2011, 18:54
thanks Karishma ...my doubt is exactly the same thing but for 16 ..how come 42 is correct if n and 16 have a gcf of 4
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Re: The greatest common factor of 16 and the positive integer n [#permalink]  22 Feb 2011, 18:59
1
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Expert's post
ajit257 wrote:
thanks Karishma ...my doubt is exactly the same thing but for 16 ..how come 42 is correct if n and 16 have a gcf of 4

That is because 210 has only one 2. Even though n has a 4 as factor, 210 does not. Therefore GCF of n and 210 does not have 4 as a factor. Does it make sense now?
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Re: The greatest common factor of 16 and the positive integer n [#permalink]  22 Feb 2011, 19:01
so lets say if another choice was given as 6 then what could have been the ans or it would not be that close.
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Re: The greatest common factor of 16 and the positive integer n [#permalink]  22 Feb 2011, 19:09
Expert's post
ajit257 wrote:
so lets say if another choice was given as 6 then what could have been the ans or it would not be that close.

Then there would have been two correct options: 6 and 42. Either can be the GCF of n and 210 depending on what exactly n is.
GMAT never has 2 correct options and hence such a scenario is not possible. Only one of 6 and 42 would be in the answer choices.
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The greatest common factor of 16 and the positive integer n [#permalink]  25 Feb 2011, 07:21
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
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Re: GCF [#permalink]  25 Feb 2011, 07:32
1
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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
1
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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, 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: The greatest common factor of 16 and the positive integer n [#permalink]  23 Jan 2014, 13:09
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Re: GCF [#permalink]  21 Apr 2014, 07:35
Bunuel wrote:
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).

Hi Bunnel,

I have resolved this till n is a multiple of 2^2*3=12

why we are not considering 5 but we are considering 7.

Thanks
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Re: GCF [#permalink]  21 Apr 2014, 07:57
Expert's post
PathFinder007 wrote:
Bunuel wrote:
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).

Hi Bunnel,

I have resolved this till n is a multiple of 2^2*3=12

why we are not considering 5 but we are considering 7.

Thanks

Notice that the GCF of 45 (a multiple of 5) and n is 3 (not a multiple of 5). This means that n itself cannot be a multiple of 5.

As for 7: we know for sure that 2 and 3 are factors of n and 5 is not a factor of n. We know nothing about its other primes, so any prime greater than 5 theoretically can be a factor of n.
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Re: The greatest common factor of 16 and the positive integer n [#permalink]  03 Jun 2014, 05:56
1st pair (n and 16) for whom the GCF is 4

GCF=4=2^2
16=2^4
Since GCF contains the lowest powers of all the common prime factors
it can be deducted that n must contain 2^2

2nd pair (n and 45)for whom the GCF is 3
GCF=3=3^1
45=3^(2 ) x 5^1
Since GCF contains the lowest powers of all the common prime factors
it can be deducted that n must contain 3^1 and must not contain 5^1

3rd Pair (n and 210)
210=2 x 3x 5 x 7
n=must contain 2^2,must contain 3^1,may contain 7,must not contain 5
Therefore n could be either=(2^2 x 3^1=12)or (2^2 x 3^1 x 7^1=84)

if n=12 then GCF of 12 and 210 is 2 x 3=6
if n=84 then GCF of 84 and 210 is 2 x 3 x 7=42

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Re: The greatest common factor of 16 and the positive integer n [#permalink]  03 Jun 2014, 23:43
16........ n .......................... n ....... 45

GCF = 4 ................................. GCF = 3

So n may be either 4 * 3 = 12 or a multiple of 12 to satisfy the GCF values given.

210 = 7 * 2 * 5 * 3

2, 3 & 5 are associated with 16 & 45. Including them in calculations will disturb the already provided GCF values

Only 7 stands out.

So 12 * 7 = 84

GCF of 84 & 210 = 42

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Re: The greatest common factor of 16 and the positive integer n   [#permalink] 03 Jun 2014, 23:43
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