Greatest common factors

j_shreyans 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

The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3.
Draw a VENN DIGRAM showing where the prime factors of n, 16 and 45 overlap:


The diagram implies the following:
Since n and 16 have only two 2's in common, the prime-factorization of n includes exactly two 2's.
Since n and 45 have only a 3 in common, the prime-factorization of n includes exactly one 3 and no 5's.
Thus:
n = 2*2*3*k, where k is not a multiple of 2, 3, or 5.

Since n is not a multiple of 5, the GCF of n and 210 cannot be a multiple of 5.
Eliminate C and E.
Since n = 2*2*3*k and 210 = 2*3*5*7, the GCF of n and 210 must be a multiple of 6, as indicated by the factors in red.
Eliminate A and B.

The correct answer is D.

j_shreyans wrote:Guys ,

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

OAD

Since n and 16 have GCD = 4 therefore n will be odd multiple of 4
Since n and 45 have GCD = 3 therefore n will be a multiple of 3 but not a multiple of 9 and 5

i.e. n is multiple of 4 and 3 but not a multiple of 9 and 5

therefore GCD of n and 210 will have factors 2 and 3 certainly and hence the result must be a multiple of 2 and 3 [eliminating option A and B] BUT MUST NOT be multiple of 5 and 9 [eliminating option C and E]

Only option Left is Option D

Answer: Option D

Hi ,

Is there any other way to solve this type of questions?

j_shreyans wrote:Hi ,

Is there any other way to solve this type of questions?

I think you have the best ways to look at the Questions mentioned here. The most important part for solving the questions of Number properties is to Observe the numbers and work within the constraints.

In case you want to take an alternate way then you might be mechanical while dealing with the information which is not expected out of GMAT test takers.

So keep your eyes open for the clues that questions offers and use them effectively.