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.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
