Problem Solving

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


[spoiler]Answer: D[/spoiler]

N must have 4 & 3 proved from the two statements.

So N is a compound of 12.

Greatest Common Factor (GCF) of
12 & 210 -> 6
24 & 210 -> Not possible as GCF of 24 & 16 is 8, not 4
36 & 210 -> Not possible as GCF of 36 and 45 is 9, not 3
48 & 210 - Not possible as GCF of 16 & 48 is 16, not 4
60 & 210 - Not possible as GCF of 45 & 60 is 15, not 3
72 and 210 - Not possible as GCF of 16 & 72 is 8, not 4
84 and 210 - GCF 42
96 and 210 - Not possible as GCF of 16 & 96 is 16, not 4

That's enough for calculation, as after this all the compound forms of 12 is multiple of previous numbers.

sparkle6 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


[spoiler]Answer: D[/spoiler]

GCD of 16 and n = 4
Let n be 4x.
It means x cannot be even. ____(i)

Since GCD of 45 and n is 3 and 45=3*3*5.
x can have only one 3 as a factor and cannot have 5 as a factor. _____ (ii)

Now 210=2*5*3*7
and n=2*2*3*(some number, say y) (from i and ii)

GCD of 210 and n can only 2,5,3 and 7 as a factor. But 5 is not possible, so we are left with only one possiblity from the given options, i.e. 2*3*7 = 42