In the list of numbers 41, 43, 47, 53, 61, 71, 73, 81, which number is the odd one out?

Introduction / Context:
This question asks you to identify the odd man out in a set of numbers. Typically, such problems are based on number properties such as primality, divisibility, or belonging to some specific mathematical pattern. Here, most of the numbers form a natural group based on being prime numbers, and one does not fit that group. Recognizing prime numbers quickly is a key skill for this kind of problem.

Given Data / Assumptions:

• The numbers given are 41, 43, 47, 53, 61, 71, 73, and 81.
• We must choose the single number that does not share the key property common to the others.
• All numbers are positive integers greater than 1.

Concept / Approach:
The natural property to check here is whether each number is prime. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. By testing each number for divisibility by small primes such as 2, 3, 5, 7, and 11, we can determine which numbers are prime and identify if there is one that is composite. The composite number among mostly prime numbers will be the odd man out.

Step-by-Step Solution:
Check 41: It is not divisible by 2, 3, 5, or 7. Thus 41 is a prime number. Check 43: It is not divisible by 2, 3, 5, or 7. Thus 43 is a prime number. Check 47: It is not divisible by 2, 3, 5, or 7. Thus 47 is a prime number. Check 53: It is not divisible by 2, 3, 5, or 7. Thus 53 is a prime number. Check 61: It is not divisible by 2, 3, 5, or 7. Thus 61 is a prime number. Check 71: It is not divisible by 2, 3, 5, or 7. Thus 71 is a prime number. Check 73: It is not divisible by 2, 3, 5, or 7. Thus 73 is a prime number. Check 81: 81 = 9 * 9 = 3^4, so it is clearly composite and not prime. Since all the other numbers are prime and 81 is composite, 81 is the odd man out.

Verification / Alternative Check:
To verify our conclusion, we can focus on 81 and note that it is an obvious power of 3. Being equal to 3 * 3 * 3 * 3, it has divisors 1, 3, 9, 27, and 81. In contrast, for a number like 41, any factor other than 1 and 41 would have to be less than or equal to the square root of 41, which is slightly above 6, but 41 is not divisible by 2, 3, 5, or any other integer in that range. The same reasoning works for the other listed primes. Thus our identification of 81 as the only composite number in the group is correct.

Why Other Options Are Wrong:

• Option 41: This is a prime number and shares the same property as most of the numbers in the list.
• Option 61: This is also a prime number and does not stand out in terms of primality.
• Option 71: Another prime, so it matches the majority pattern.
• Option 73: This is prime as well and is not different in the sense required by the question.

Common Pitfalls:
A common error is to overcomplicate the pattern search and look for hidden relationships between consecutive numbers, such as their differences or sums, while missing the simple property of primality. Another pitfall is to misjudge primality by not testing divisibility properly, for example assuming that a number ending with 1 is always prime. Remember that a methodical check for small prime divisors is usually enough for numbers in this range.

Final Answer:
The odd man out in the list is 81.