GCF

akshatgupta87 wrote:Q.) 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)4
c)30
d)42
e)70

Can someone explain, how to tackle this type of question.

IMO D.

GCF ( 16, n) = 4 i.e n is max multiple of 4
GCF (45, n) = 3 i.e n is also multiple of 3 but not multiple of 5.
GCF( 210, n) = ( 210=2*3*5*7 , n=2*2*3*...) GCF = 2 * 3* 7 ( but not 5 as per 2nd statement) = 42

akshatgupta87 wrote:Can someone explain, how to tackle this type of question.

Target2009 wrote: GCF ( 16, n) = 4 i.e n is max multiple of 4
GCF (45, n) = 3 i.e n is also multiple of 3 but not multiple of 5.
GCF( 210, n) = ( 210=2*3*5*7 , n=2*2*3*...) GCF = 2 * 3* 7 ( but not 5 as per 2nd statement) = 42

Hi there!

Target2009´s reasoning is very good, but let me "structure" the whole thing a bit more, for us to be able to obtain everything the info given let us conclude about the positive integer n !

01. From GCF(16=2^4,n)=4 we may conclude that n=4M, where M is a positive ODD integer (why?);
02. From GCF(45=3^2*5,4M)=3 we may conclude that M=3K, where K is positive ODD integer (because of M) that is not a multiple of 3 nor a multiple of 5 (why?) ;

Combining the info above, we know that n=12K, where K is a positive ODD integer that is not a multiple of 3 nor a multiple of 5... what about GCF(210=2*3*5*7,n) ?

GCF(210=2*3*5*7,n)= GCF(2*3*5*7,2^2*3*K where K...) = 2*3*W, where W is a positive integer equal to 1 or to 7, therefore there are only two possibilities for GCF(210,n), they are: 6 and 42.

I hope you enjoy this discussion.

Regards,
Fabio.