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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]

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New post 22 Feb 2011, 17:01
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D
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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
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Re: The greatest common factor of 16 and the positive integer n  [#permalink]

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New post 22 Feb 2011, 18:38
11
3
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)

Answer (D)

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]

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New post Updated on: 22 Feb 2011, 18:51
5
1
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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Originally posted by AmrithS on 22 Feb 2011, 18:36.
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]

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New post 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]

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New post 22 Feb 2011, 18:59
1
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]

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New post 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]

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New post 22 Feb 2011, 19:09
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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Re: GCF  [#permalink]

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New post 25 Feb 2011, 07:32
4
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).

Answer: D.
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Re: GCF  [#permalink]

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New post 25 Feb 2011, 08:34
3
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]

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New post 04 Nov 2012, 05:43
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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]

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New post 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. :o :o
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Re: Question on GCF  [#permalink]

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New post 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: GCF  [#permalink]

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New post 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).

Answer: D.



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]

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New post 21 Apr 2014, 07:57
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).

Answer: D.



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]

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New post 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

42 is the correct answer
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Re: The greatest common factor of 16 and the positive integer n  [#permalink]

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New post 03 Jun 2014, 23:43
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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

Answer = D
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Re: The greatest common factor of 16 and the positive integer n  [#permalink]

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New post 04 Aug 2015, 03:16
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?

A. 3
B. 14
C. 30
D. 42
E. 70


:twisted:
GCF (n, 16) = 4
GCF (n, 45) = 3

so n only contains 4 and 3 no 5

210 = 2*3*5*7

lowest GCF is 6
possible GCF in answer choice is 42


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

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New post 29 Jan 2017, 03:20
This is an excellent Question.
Here is what i did in this question -->


GCD(N,16)=4
Hence N=2^2*k
where 2 is not the prime factor of k.
GCD(N,45)=3
N=3*m
where 3 and 5 are not prime factors of m.

Combing the two equations above -->
N=2^2*3*x
where 2,3,5 are not prime factors of x.


Now 210=2*3*5*7
Hence GCD (N,210)=2*3*common factors among x and 7
So GCD must be a multiple of 6.
But wait a second.
There are two options which are multiples of 6.
Notice for GCD to be 30 => x must have 5 as its prime.
But x cannot have 5 as its prime.
Hence 30 is out too.

So the only viable choice is 42.
As it turns out if x=7 => GCD(N,210)=42

Hence D.

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Re: The greatest common factor of 16 and the positive integer n  [#permalink]

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New post 04 Jun 2017, 18:07
1
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?

A. 3
B. 14
C. 30
D. 42
E. 70


This problem works best if you break it down into pieces with Venn diagrams.

16 and n have a GCF of 4 = 2^2. That means n has at least two 2's as factors.

45 and n share one factor: 3. That means n has at least one 3 and two 2's (3*2^2).

The next step is to break down 210 into 2*3*7*5. 5 is not a possible factor of n because it would've been a GCF with 45 and n. But 7 could be a common divisor because it's not expressly forbidden anywhere. So the GCF in this case is 2*3*7 = 42.
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Re: The greatest common factor of 16 and the positive integer n  [#permalink]

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New post 01 Oct 2018, 15:26
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?

A. 3
B. 14
C. 30
D. 42
E. 70

\(?\,\,\,:\,\,\,GCF\left( {n\,,2 \cdot 3 \cdot 5 \cdot 7} \right)\,\,\underline {{\rm{could}}\,\,{\rm{be}}}\)

\(n \ge 1\,\,\,{\mathop{\rm int}}\)

\(GCF\left( {{2^4},n} \right) = {2^2}\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
{n \over {{2^2}}} = {\mathop{\rm int}} \hfill \cr
{n \over {{2^{\, \ge \,3}}}} \ne {\mathop{\rm int}} \hfill \cr} \right.\)

\(GCF\left( {{3^2} \cdot 5,n} \right) = 3\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
{n \over 3} = {\mathop{\rm int}} \hfill \cr
{n \over {{3^{\, \ge \,2}}}} \ne {\mathop{\rm int}} \,\,\,\,\,;\,\,\,{n \over 5} \ne {\mathop{\rm int}} \,\, \hfill \cr} \right.\)

\(? = {2^1} \cdot {3^1} \cdot {7^{0\,{\rm{or}}\,1}}\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{alternatives}}\,!} \,\,\,\,42\,\,\,\,\,\,\left( D \right)\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: The greatest common factor of 16 and the positive integer n &nbs [#permalink] 01 Oct 2018, 15:26

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