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

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

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03 Oct 2018, 17:43
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

If the greatest common factor (GCF) of 16 and n is 4, n could be 4, 12, 20, 28, 36, etc. In other words, n is an odd multiple of 4.

Since the GCF of 45 and n is 3, and since 45 and 4 have no common factor other than 1, n must be a multiple of 3 x 4 = 12. However, since n is an odd multiple of 4, n actually has to be an odd multiple of 12 also.

If n = 12, we see that GCF(45, 12) = 3 and GCF(12, 210) = 6. However, 6 is not one of the choices.

If n = 36, we see that GCF(45, 36) = 9, but GCF(45, n) is supposed to be 3. So n can’t be 36.

If n = 60, we see that GCF(45, 60) = 15, but GCF(45, n) is supposed to be 3. So n can’t be 60.

If n = 84, we see that GCF(45, 84) = 3 and GCF(84, 210) = 42. We see that 42 is one of the choices.

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

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04 Oct 2018, 19:04
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.

Hi Karishma,

Can you please elaborate on …….so GCF of n and 210 could be 6 ?I cant figure out how 6 is the factor of n and 210.How can we be certain that 9 is a common factor for n and 210?

Thank you.
Re: The greatest common factor of 16 and the positive integer n   [#permalink] 04 Oct 2018, 19:04

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

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