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The greatest common factor of 16 and the positive integer n
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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
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Re: The greatest common factor of 16 and the positive integer n
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22 Feb 2011, 18:38
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
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Updated on: 22 Feb 2011, 18:51
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.
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Re: The greatest common factor of 16 and the positive integer n
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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
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22 Feb 2011, 18:59
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
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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
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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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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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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
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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
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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
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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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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=12why we are not considering 5 but we are considering 7. Thanks



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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=12why 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
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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
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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 Answer = D
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Re: The greatest common factor of 16 and the positive integer n
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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 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 Gib kudos



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The greatest common factor of 16 and the positive integer n
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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
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04 Jun 2017, 18:07
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
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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
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