MGMAT Q Bank

[mridul12]

What is the greatest common factor of positive integers a and b?

(1) a = b + 4

(2) b/4 is an integer


What is an efficient way to solve this problem or this type of problem

[anadi]

1) A = B + 4..........not sufficient

2) B is divisible by 4, then say B = 4C where C is an integer. (Insufficient alone)

Now, B=4C and A = 4(C+1). For positive integers , C and C+1 will have only 1 as a comon factor. So GCF of A and B is 4. Sufficient.

[StaceyKoprince]

You might also try picking some real numbers and seeing whether you can come up with multiple answers, in which case a statement isn't sufficient.

(1) a = b+4
let's say a=6 and b=2. GCF is 2.
let's say a=8 and b=4. GCF is 4.
Insufficient. Elim A and D.

(2) b/4 is an integer
tells me nothing about A, by itself. Insufficient. Elim B.

(1) AND (2) now I can use theory to solve this
(2) tells me b is divisible by 4 - so 4 is a factor of b. One of my divisibility rules (found in the divisibility and prime chapter of our Number Properties strategy guide) is that when you add or subtract numbers, the sum or difference will share any common factors. So b has 4 as a factor and 4 has 4 as a factor, therefore, 4 is also a factor of a.

But is it the greatest common factor? Another rule is that if two numbers (a and b) share a factor and the difference between those numbers is equal to that factor, then that factor is the GCF. a and b share 4 as a factor, and the two numbers are 4 units apart (from statement 1)... so 4 is the GCF.