In this divisibility question, we can either TEST CASES or use DIVISIBILITY THEORY.

If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

Testing Cases:
Case 1:
b = 6
a = 2(6) + 6 = 18
the GCF of 6 and 18 is 6 --> yes.

Case 2:
b = 12
a = 2(12) + 6 = 30
the GCF of 12 and 30 is 6 --> yes.

We can keep testing, but we'll get a "yes" answer every time.

Divisibility theory:
Since b is divisible by 6, imagine b = 6n, where n is some integer.
Thus, a = 2(6n) + 6
We could factor: a = 6(2n + 1)
Since 2n + 1 can never share factors with n (other than 1 itself), a cannot share factors with b other than 6. Thus, we know that 6 must be the greatest common factor. Sufficient.

(2) a = 3b

Testing Cases:
Case 1:
b = 6
a = 18
the GCF of 6 and 18 is 6 --> yes.

Case 2:
b = 12
a = 36
the GCF of 12 and 36 is 12 --> no.

We got a "yes" and a "no" answer to the question, so this is insufficient.

Divisibility theory:

If a is 3 times as great as b, it will contain b as a factor. That means that the value of b itself will always be the greatest common factor of a and b. Maybe that GCF is 6, or maybe it's some other multiple of 6. Insufficient.

The answer is A.