Does positive integer b have a factor n such that 1 < n < b ?

Step 1: Analyze Question Stem

• We need to find out if the positive integer b has a factor n that lie between 1 and b.
• Now, there can be two scenarios:

o Case 1: b has a factor that lies between 1 and b. This would mean that b is a composite number. For example: If b= 6 then b has factors 2 and 3 between 1 and 6.
o Case 2: b does not have a factor between 1 and b. This would mean that b is a prime number. For example: b = 5 does not have any factor between 1 and 5.

So, the question is indirectly asking us to find out, if b is a composite number or a prime number?

Step 2: Analyze individual statements

Statement 1: b = 2k, where k is an integer greater than 1.

• If k is an integer greater than 1.

o Then b is an even number greater than 2.

 Which means b is definitely a composite number.

Hence, statement 1 is sufficient and we can eliminate answer options B, C, and E.

Statement 2: k is a factor of b, where k is an integer greater than 1.

• If k is factor of b and k is an integer, greater than 1, then there can be 2 possibilities.

o Case 1: Say b = 6

 Then k can be 2 or 3 or 6 and b is a composite number.
o Case 2: Say b = 5

 Then k will be equal to 5 and b is a prime number.

Since, we are getting two different answers, statement 2 is not sufficient and the correct answer is Option A.
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