How many composite numbers lie between 67 and 101 (excluding both 67 and 101)?

Introduction / Context:
This problem belongs to the number system and primes versus composites topic. You are asked to count composite numbers in a given interval. A composite number is a positive integer greater than 1 that is not prime, meaning it has at least one positive divisor other than 1 and itself. Carefully distinguishing prime numbers from composite numbers in a range is a common and important skill in many competitive exams.

Given Data / Assumptions:
- The range under consideration is from 67 to 101, but 67 and 101 themselves are excluded because the question says between these numbers.
- Therefore, the actual numbers to consider are from 68 up to 100 inclusive.
- We must count how many of these numbers are composite.
- A composite number is greater than 1 and not prime.

Concept / Approach:
The easiest way is to count how many integers are in the range from 68 to 100 and then subtract the number of primes in that range. All remaining numbers greater than 1 will be composite. Primes in this region can be identified using basic divisibility tests by small primes such as 2, 3, 5, 7, and 11. This method is simpler and less error prone than checking each number directly for composite status.

Step-by-Step Solution:
Step 1: Determine how many integers lie between 68 and 100 inclusive.Step 2: The count is 100 - 68 + 1 = 33 numbers.Step 3: Next, identify prime numbers between 68 and 100.Step 4: Check known primes in this region: 71, 73, 79, 83, 89, and 97 are prime numbers.Step 5: Confirm that the other numbers in the range are not prime. For example, 69 is divisible by 3, 70 by 2 and 5, 77 by 7 and 11, 91 by 7 and 13, and 93 by 3 and 31.Step 6: Therefore, there are 6 prime numbers in the range 68 to 100 inclusive.Step 7: All remaining numbers greater than 1 in this interval must be composite.Step 8: The number of composite numbers is 33 total numbers minus 6 primes, which equals 27.

Verification / Alternative check:
You can verify by listing the primes explicitly: 71, 73, 79, 83, 89, and 97. Every other number from 68 to 100 either ends in an even digit or 5, or passes a divisibility test for 3, 7, or 11. Therefore, none of them are prime. Since the total count of numbers is 33 and we have exactly 6 primes, the count of composites must be 27. This method matches our previous calculation and strengthens confidence in the result.

Why Other Options Are Wrong:
Options such as 24, 26, 23, or 25 correspond to incorrect counts that likely come from misidentifying some primes as composites or vice versa, or from misunderstanding the word between and mistakenly including 67 or 101. If you wrongly include 67 or 101 or miscount primes in the interval, you will get one of these incorrect values instead of 27.

Common Pitfalls:
A frequent mistake is to forget that the endpoints 67 and 101 are not part of the count. Another typical error is to overlook primes such as 79 or 83 due to insufficient divisibility checking and treat them as composite. Ensuring a systematic test for each candidate prime and counting the interval correctly avoids these issues.

Final Answer:
The number of composite numbers between 67 and 101 is 27.