bubbliiiiiiii 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
First watch this free video about finding the Greatest Common Factors:
https://www.gmatprepnow.com/module/gmat- ... /video/833
Now, we can use ELIMINATION to find the correct answer.
Goal: Find GCF of 210 and n.
210 = (2)(3)(5)(7)
The greatest common factor of 16 and the positive integer n is 4
16 =
(2)(2)(2)(2)
4 =
(2)(2)
So, we know for certain that the PRIME FACTORIZATION of n has TWO 2's (and no more than TWO)
In other words, n =
(2)(2)(?)(?)...
Since 210 =
(2)(3)(5)(7), we can see that 210 and n both share ONE
2.
So, the GCF of 210 and n will be divisible by
2.
We can ELIMINATE A, since 3 is not divisible by
2.
The greatest common factor of n and 45 is 3
45 = (
3)(3)(5)
3 =
3
So, we know for certain that the PRIME FACTORIZATION of n has ONE
3 (and no more than ONE)
In other words, n = (
3)(?)(?)(?)...
Since 210 = (2)(
3)(5)(7), we can see that 210 and n both share ONE 3.
So, the GCF of 210 and n will be divisible by
3.
We can ELIMINATE B and E, since they are not divisible by
3.
This leaves C (30) and D (42).
Finally, notice that, since the GCF of n and 45 is 3, we can be certain that n does NOT have a 5 in its prime factorization. Otherwise, the GCF of n and 45 would be 15.
This means that n is NOT divisible by 5, which also means the GCF of 210 and n will NOT be divisible by 5
So we can ELIMINATE C.
This leaves us with
D, the correct answer.
Cheers,
Brent
