pkw209 wrote:The greatest common factor of 16 and the positive integer N is 4. The GCF of N and 45 is 3. Which of the following could be the GCF of N and 210?

a) 3

b) 14

c) 30

d) 42

e) 70

Answer is D

We know that when N is written as a product of prime numbers, the power of 2 will be 2, since GCF(16,N)=4
(N must be a multiple of 4, but not of 8, as the GCF is 4 )

Since GCF(N,45) is 3 and 45= 3^2 x 5, the power of 3 in the prime factorization of N is 3^1 and that of 5 is 5^0

Thus N = 2^2 x 3 x 5^0 x ...
210 = 2 x 3 x 5 x 7

Thus GCF(N, 210) = 2 X 3 x 7^x , where x is either 0 or 1

Therefore GCF(N, 210) is either 6 or 6 x 7= 42

pkw209 wrote:The greatest common factor of 16 and the positive integer N is 4. The GCF of N and 45 is 3. Which of the following could be the GCF of N and 210?
a) 3 b) 14 c) 30 d) 42 e) 70
Answer is D

GCF (16,N) is 4 so N is not 2*4
GCF (45,3) is 3 so N is not 3*5, 3*3
N = 2*2*3*k where k is a prime no (k is is not 2,3,5)
210 = 2*3*5*7
Greatest CF could be 6 or multiple of 6
3 is not the greatest
14 is not a multiple of 6
30 is but k is not 5
42 possible as k can be 7
70 is not a multiple of 6