BTGmoderatorDC 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
OA D
Source: Manhattan Prep
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 number.
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 number, n actually has to be a product of 12 and an odd number 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 GCF(16, n) = 4, we see that 4 is a factor of n, but 8 is not (otherwise, the GCF would have been at least 8).
Since GCF(n, 45) = 3, we see that 3 is a factor of n, but 9 and 5 are not.
In light of these facts, let's test each answer choice:
GCF(n, 210) = 3
Since n is divisible by 2 and 3, n is divisible by 6. Likewise, 210 is also divisible by 6. So, GCF(n, 210) is at least 6; this cannot be the correct answer choice.
GCF(n, 210) = 14
As explained in the previous answer choice, both n and 210 are divisible by 6; therefore, GCF(n, 210) must also be divisible by 6. This cannot be the correct answer choice.
GCF(n, 210) = 30
Since GCF(n, 45) = 3, 5 is not a factor of n. Thus, GCF of n and 210 cannot be divisible by 5; this cannot be the correct answer choice.
GCF(n, 210) = 42
Since 42 is divisible by 6, the GCF of n and 210 could be 42 if n was divisible by 7.
GCF(n, 210) = 70
Just as in explained in answer choice C, GCF of n and 210 cannot be divisible by 5; thus, this cannot be the correct answer choice.
The only possible answer choice is D.
Answer: D
