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

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22 Feb 2011, 18:01

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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?

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.
_________________

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?
_________________

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.
_________________

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

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25 Feb 2011, 08:21

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?

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).

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

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?

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
_________________

Exhaust your body, proceed your mind, cultivate your soul.

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.

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

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23 Jan 2014, 14:09

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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

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.

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.
_________________

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

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03 Jun 2014, 06:56

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

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

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04 Aug 2015, 04:16

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

gmatclubot

Re: The greatest common factor of 16 and the positive integer n
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04 Aug 2015, 04:16

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