ajit257 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
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 an odd multiple of 4.
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 an odd multiple of 4, n actually has to be an odd multiple of 12 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.
Answer: D
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