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by bubbliiiiiiii » Fri Aug 28, 2015 3:40 am
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

OA D
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by GMATGuruNY » Fri Aug 28, 2015 4:43 am
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 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3.
Draw a VENN DIGRAM showing where the prime factors of n, 16 and 45 overlap:
Image

The diagram implies the following:
Since n and 16 have only two 2's in common, the prime-factorization of n includes exactly two 2's.
Since n and 45 have only a 3 in common, the prime-factorization of n includes exactly one 3 and no 5's.
Thus:
n = 2*2*3*k, where k is not a multiple of 2, 3, or 5.

Since n is not a multiple of 5, the GCF of n and 210 cannot be a multiple of 5.
Eliminate C and E.
Since n = 2*2*3*k and 210 = 2*3*5*7, the GCF of n and 210 must be a multiple of 6, as indicated by the factors in red.
Eliminate A and B.

The correct answer is D.
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by Brent@GMATPrepNow » Sat Aug 29, 2015 9:39 am
bubbliiiiiiii 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
First watch this free video about finding the Greatest Common Factors: https://www.gmatprepnow.com/module/gmat- ... /video/833

Now, we can use ELIMINATION to find the correct answer.

Goal: Find GCF of 210 and n.
210 = (2)(3)(5)(7)

The greatest common factor of 16 and the positive integer n is 4
16 = (2)(2)(2)(2)
4 = (2)(2)
So, we know for certain that the PRIME FACTORIZATION of n has TWO 2's (and no more than TWO)
In other words, n = (2)(2)(?)(?)...
Since 210 = (2)(3)(5)(7), we can see that 210 and n both share ONE 2.
So, the GCF of 210 and n will be divisible by 2.
We can ELIMINATE A, since 3 is not divisible by 2.


The greatest common factor of n and 45 is 3
45 = (3)(3)(5)
3 = 3
So, we know for certain that the PRIME FACTORIZATION of n has ONE 3 (and no more than ONE)
In other words, n = (3)(?)(?)(?)...
Since 210 = (2)(3)(5)(7), we can see that 210 and n both share ONE 3.
So, the GCF of 210 and n will be divisible by 3.
We can ELIMINATE B and E, since they are not divisible by 3.

This leaves C (30) and D (42).

Finally, notice that, since the GCF of n and 45 is 3, we can be certain that n does NOT have a 5 in its prime factorization. Otherwise, the GCF of n and 45 would be 15.
This means that n is NOT divisible by 5, which also means the GCF of 210 and n will NOT be divisible by 5
So we can ELIMINATE C.

This leaves us with D, the correct answer.

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Sat Aug 29, 2015 8:37 pm
Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.


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
===> If A=aG, B=bG (a,b = relative prime numbers==> common factor is only 1), then L=abG.
In other words, from GCF(n,16)=4, n=4*m(m can not have 4 as factor since 16=4*4, and we need 4 and m to be relative prime numbers)
And since GCF(n,45)=3 we have n=3*k(k can not have 15 as factor 45 = 3*15, and we need 15 and k to be relative prime numbers)

210=2*3*5*7, n=4*3*s and s can't have 4 and 15 as factor. Since 5 should not be included, 30 and 70 is out.
210=6*35, and since GCF(n,210) must have 6, we can excluded 3 and 14. Therefore 42 is the answer.

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