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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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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?

A. 3
B. 14
C. 30
D. 42
E. 70
[Reveal] Spoiler: OA

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17 Sep 2008, 01:09
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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?

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

7*3*2 = 42

D.

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18 May 2012, 07:58
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mila84 wrote:
Spidy001 wrote:
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).

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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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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.

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16 Sep 2008, 08:03
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?

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

210 = 5*2*3*7 from the choices 3 wins

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16 Sep 2008, 17:13
42

5 cannot be a factor of n (between n and 45 common factor is only 3 not 5)

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23 Sep 2011, 02:43
I believe it should be 3 based on the given data. "N" can be 12 and satisfy the two conditions(2,2,3).

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23 Sep 2011, 02:58
The questions states "could be" & not "must be".

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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25 Sep 2011, 14:10
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.

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18 May 2012, 04:11
Spidy001 wrote:
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!!

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

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01 Aug 2014, 23:49
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
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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
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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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The greatest common factor of 16 and the positive integer n   [#permalink] 22 Mar 2017, 08:51
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