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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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21 Dec 2008, 05:18
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: the-greatest-common-factor-of-16-and-the-positive-integer-n-109870.html
[Reveal] Spoiler: OA

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21 Dec 2008, 12:24
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I would say 42.

16 = 2^4. 16 and n have GCF as 2^2. That means, n could have only 2 twos as the factor.

45 = 5*3^2. 45 and have GCF as 3. That means, n will not have 5 as factor and only one 3 as the factor.

210 = 2*3*5*7 and since n does not have 5 as factor and n has 2^2 and 3 as factors, n could as well have 7 as a factor. Hence, 2*3*7 would be the greatest common factor.
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21 Dec 2008, 12:35
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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

N= 4X, N=3Y THUS N = 12A

210 = 5*2*3*7

N HAS NO FACTOR 5 , US 42 COULD BE
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21 Dec 2008, 22:41
krishan 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

210 = 5*2*3*7

N = 4x * 3 y = 2^2*3xy

6 must be the minimum common factor between n and 210.
Answer must be divisible by 6

only 30 and 42 left.
Maximum value could be 42.

If answer choice has 210 then will chose 210
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22 Dec 2008, 08:55
im confused.

i got up to the point where 6 = lowest common factor.

then i understand that the answer has to be divisible by 6, therefore 30 or 42. why not 30?
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22 Dec 2008, 11:53
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sidrah wrote:
im confused.

i got up to the point where 6 = lowest common factor.

then i understand that the answer has to be divisible by 6, therefore 30 or 42. why not 30?

"Which of the following could be the greatest common factor of n and 210?"
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22 Dec 2008, 14:02
30 cannot be an answer because N cannot be divisible by 5 because:

45 has factors of 5x3x3. If N had a factor of 5, then 45 and Ns greatest common factor would be 5, not 3
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22 Dec 2008, 18:01
yay i geddit! =)

was confused because simply coz 42 is bigger than 30 does not make it the right answer ... that would be assuming both are factors of n and 210.

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23 Dec 2008, 12:35
krishan 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

agree with 42.

16 = 2x2x2x2
45 = 3x3x5
n = 4 x 3 x k, where k is an integer
210 = 2x3x5x7

gcf of 16 and n = 4
gcf of n and 45 = 3
gcf of n and 210 = 2x3x7 (5 and 4 cannot be factors of n because 210 has only 2 as factor and n doesnot have 5 as factor) = 42

Another good question!!!!!!
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23 Dec 2008, 22:31
Since u have to find the GCF between 210 and n, go by the answer options.

first consider 70 = 7*5*2 . none of this coincides with conditions given. i.e GCF 4 and 3 is not obtained if u take 7*5*2.

move on to next option 42=7*2*3*k
take k=1; one condition is matched.
take k=2; both the conditions are matched . So answer is multiple of 42.
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24 May 2011, 05:27
FOR ME:
NUMBER PRIME FACTORING N CANT BE INCLUDED IN N
16 2^4 2^2*X 2 OR 2^2
45 3^3*5 3*N 3 OR 5

210 PRIME FACTORIZATION IS 2*5*3*7

2, 5,3 CANT BE AND ONLY 7 HENCE:

7*2^2*3
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01 Apr 2014, 13:35
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Re: The greatest common factor of 16 and the positive integer n [#permalink]

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01 Apr 2014, 16:01
Expert's post
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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).

OPEN DISCUSSION OF THIS QUESTION IS HERE: the-greatest-common-factor-of-16-and-the-positive-integer-n-109870.html
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Re: The greatest common factor of 16 and the positive integer n [#permalink]

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19 Apr 2014, 08:26
Not a proper but a random approach:

GCD(n,16) = 4

n--------------2^4

n must at least be a multiple of 4

GCD(n,45) = 3

n----------------------5*3^2

n must have one '3' and shouldn't have 5

Applying above restrictions calculating

Max GCD(n,210)

n-------------------------2*3*5*7

N has one 3 and one 4 so

n= 3*2*2

n cannot have 5 but can have 7 so

n=2*2*3*7 --------------------2*3*5*7

Max(GCD) = 42
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Re: The greatest common factor of 16 and the positive integer n   [#permalink] 19 Apr 2014, 08:26
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