This is a confusingly-worded question. It specifies that b is a positive integer. Then it asks whether b has a factor that is greater than 1 but less than b itself. Can you think of a number for which this would be true?
For example, if b = 10, then it has a factor 5, and that factor is between 1 and 10. (It also has a factor 2, which is between 1 and 10.)
Can you think of a number for which this description would not be true?
For example, if b = 7, then its only factors are 1 and 7. It has no factor that is greater than 1 but less than 7.
Essentially, the question is asking whether b is a prime number, such as 7, or a composite number (a composite number is essentially* a non-prime positive integer), such as 10.
(*The number 1 is the only positive number that is neither prime nor composite. It has its own unique category. But you can ignore this fact from the point of view of the GMAT!)
(1) SUFFICIENT: Test some real numbers to see what's going on. The variable k is an integer greater than 1.
Case 1: k = 2. In this case, b = 2(2) = 4. If b = 4, then it is not a prime number, so Yes, it has at least one factor that is between 1 and itself. (In this case, that factor is 2.)
Case 2: k = 3. In this case, b = 2(3) = 6. If b = 6, then it is not a prime number, so Yes, it has at least one factor that is between 1 and itself. (In this case, it has factors 2 and 3.)
Can you think of a case that will give you a No answer?
It's not possible. k has to be 2 or greater. And to get b, you have to multiply k by 2. So b has to be at least 4 and b has to be a multiple of 2. The only multiple of 2 that is prime is 2 itself. Every multiple of 2 that is greater than 2 (e.g., 4, 6, 8, ?) has 2 itself as a factor.
This statement provides a definitive answer to the question: Yes, b always has a factor between 1 and b itself. (Or: Yes, b is always a composite number.)
(2) INSUFFICIENT: Test some real numbers to see what's going on. The variable m is an integer greater than 1.
Case 1: m = 2. It's given that m is a factor of b, so b could be 4. If b = 4, then it is not a prime number, so Yes, b has at least one factor that is between 1 and itself. (In this case, that factor is 2.)
Case 2: Stick with m = 2. It's true that b could be 4, but b could also equal 2, since 2 is a factor of 2. If b = 2, then No, b does not have at least one factor that is between 1 and itself.
Since there are both Yes and No answers, this statement is not sufficient to answer the question.
Note: Every number is both a factor and a multiple of itself. For example 5 is a factor of 5 and 5 is a multiple of 5. When a problem tells you that one variable is a factor (or a multiple) of another variable, remember to include the possibility that the value is a factor (or a multiple) of itself!
The correct answer is (A): The first statement is sufficient to answer the question but the second statement is not.
