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

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16 Sep 2008, 07:44

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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

since GCF of 16 and n is 4 16 = 2*2 * 2 * 2 n = 2*2 * ...

since GCF of 45 and n is 3 45 = 5 * 3 * 3 n = 3 * ...

thus n must be 2*2 *3 *...

210 = 7 * 3 * 5 * 2

n can not be 5, (otherwise GCF of 45 and 3 would have been 15) it can be 7 though

GCF of 16 and n = 4 => n = 2^2(...) -----1 GCF of n and 45 = 3 => n =3(...) -------2

GCF of n and 210 = ?

= GCF of (2^2)*3(...) and 2*3*5*7 = 6

so GCF of n and 210 would be multiple of 6.

Answer is D as its the only possible option that is a multiple of 6.

why should the GCF of n and 210 be multiple of 6?

Thank you all!!

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).

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

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20 Apr 2013, 12:29

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mun23 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

Need easy explanation to solve it quickly

Hi, let me try to explain in simpler way:

GCF = 4 = 2^2 16 = 2^4 that means prime box of n = 2^2 , ? ? ?

GCF = 3 45 = 3^2*5 that means prime box of n = 3^1, ? ? ?

Overall prime box of n = 2^2, 3^1, ???

Now, 210 = 3*7*5*2 from above we know the prime factor and powers of n (not complete ??) therefore GCF = 3^1 * 2^1 * 7^1 (not 5 - we have seen above, but at least a 7 is possible) Thus at least a GCF of 42 is possible here

Hope this helps
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"When the going gets tough, the tough gets going!"

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

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20 Apr 2013, 21:48

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Simple explanation?

First, do a factor tree for each number.

You'll see that 5 can't be a factor of N (otherwise it would have been the highest factor between N and 45).

The highest factor that could theoretically exist between N and 210 is therefore all of the factors of 210 besides those we've ruled out. 210 is factored to 2 * 3 * 5 * 7. we've ruled out 5, so 2 * 3 * 7 = 42. Answer is D.

If the question asked "the highest factor that we KNOW exists" rather than "COULD" exist, the answer would be six since 2 and 3 are both factors of N, as well as of 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

n= 4x, n= 3y ie: n= 12z ie: 3*2*2*z , z cant be 5 or 2 or 3

The greatest common factor of 16 and the positive integer n is 4 The prime factor of n will have exactly two 2s The greatest common factor of n and 45 is 3 Exactly one 3 and exactly zero 5s Because the question states that the GCD between n and 45 is only 3. Thus, 5 cannot be a factor of n, but 7 could be a factor.

Hence, 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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09 Jul 2014, 07:20

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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n has to be a multiple of (2*2)*3 = 12 A common factor between 210= (2*3*5*7) and multiple of 12 is 2*3=6 So the G.C.F of n and 210 has to be a multiple of 6 The two choices that are multiples of 6 are 30 and 42. Bur n is not a multiple of 5 .So 30 can be ruled out and the answer is 42.
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Re: The greatest common factor of 16 and the positive integer n [#permalink]

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03 Oct 2015, 06:30

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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

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14 Mar 2016, 01:45

Here it is easy to come to the conclusion that 42 and 30 are both to be considered as the gcd must be a multiple of 6 but we need to discard 3 SO0 as 5 cannot be in N as if so the GCD of N and 45 will change hence 42 is correct So D
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Re: The greatest common factor of 16 and the positive integer n [#permalink]

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22 Mar 2017, 06:32

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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

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22 Mar 2017, 08:51

dancinggeometry 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

n's factors of 3 and 4 make it a multiple of 12 with an odd multiplier not 9, 5, 3 or 1. 7*12=84=7*3*2*2=42*2 210=7*5*3*2=42*5 42 D

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