BTGmoderatorLU wrote:Source: Manhattan Prep
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 OA is D
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 a product of 4 and an odd integer.
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 a product of 4 and an odd integer, n actually has to be a product of 12 and an odd integer 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.
Alternate Solution:
Since the GCF n and 45 is 3, n contains no factors of 5; therefore we can immediately reject answer choices C and E.
Since the GCF of n and 16 is 4, n and 210 have a common factor of 2; therefore we eliminate answer choice A.
Similarly, since the GCF of n and 45 is 3, we see that n and 210 have a common factor of 3; therefore, we eliminate answer choice B.
The only remaining choice is D.
Answer: D
