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?
If the GCF of 16 and n is 4 the maximum number of 2s that n can have is 2
(4=2^2)
If the GCF of n and 45 is 3 the maximum number of 3s that n can have is 1
so n = k*2*2*3
Lets take n and 210 = 3*7*2*5 so we know that n and 210 will share a 2 and a 3 but not a 5 ( ? , because of the 2nd condition ) , but n and 210 can share a 7 (? , conditions 1 and 2 doesn't stop us from taking 7 as a factor of n )
So n can be 3*2*7 = 42
Answer = D
There is another way to solve this
That is by eliminating all the wrong answers
3 : cannot be 3 because we know that n has 2 as a factor
14 : cannot be 14 , because n and 210 will have 3 as a common factor
30 : Cannot be 30 because , n doesn't have 5 as a factor
42 : correct
70 : Cannot be , because n , doesn't have 5 as a factor
