Among the numbers 3, 24, 186, 1008, 5040 and 20160, which one is the odd man out because its prime factorization includes a larger prime that does not appear in the others?

Introduction / Context:
This is an odd man out question that focuses on properties of prime factorization. Instead of looking for arithmetic progressions or series formulas, you need to examine which prime numbers appear in the factors of each term and identify the one that includes a different type of prime factor.

Given Data / Assumptions:
The list of numbers is:

• 3
• 24
• 186
• 1008
• 5040
• 20160

We assume that five of these numbers have prime factors drawn only from a small set of primes, while one number introduces a larger prime that does not appear in any other factorization.

Concept / Approach:
Prime factorization is the expression of a number as a product of prime numbers. To solve this question, we factor each number and observe which primes are used. If all but one number can be expressed using only the primes 2, 3, 5 and 7, then the number that needs an additional larger prime will be the odd one out.

Step-by-Step Solution:
Step 1: Factor 3. It is already a prime number, so 3 = 3.Step 2: Factor 24. We have 24 = 2^3 * 3.Step 3: Factor 186. We get 186 = 2 * 3 * 31, which introduces the prime 31.Step 4: Factor 1008. One way is 1008 = 16 * 63 = 2^4 * 3^2 * 7.Step 5: Factor 5040. A standard factorization is 5040 = 2^4 * 3^2 * 5 * 7.Step 6: Factor 20160. We can see 20160 = 4 * 5040 = 2^6 * 3^2 * 5 * 7.Step 7: Observe the primes used. The numbers 3, 24, 1008, 5040 and 20160 use only the primes 2, 3, 5 and 7. However, 186 uses the larger prime 31 in addition to 2 and 3.

Verification / Alternative check:
To verify, list the prime sets explicitly: 3 uses {3}, 24 uses {2, 3}, 1008 uses {2, 3, 7}, 5040 and 20160 use {2, 3, 5, 7}, and 186 uses {2, 3, 31}. The only number that requires a prime greater than 7 is 186. This makes it stand out from the rest of the list.

Why Other Options Are Wrong:
The numbers 24, 20160 and 5040 all have prime factors limited to 2, 3, 5 and 7. They are highly composite and share similar factor patterns. Removing any one of them as the odd man out would leave 186 still distinguished by its use of the prime 31, so they cannot be the correct choices.

Common Pitfalls:
Students may initially search for patterns in the ratios or differences between consecutive numbers and become frustrated. However, not every sequence is meant to be analyzed as a progression; some rely entirely on factorization. Remember that odd man out questions often use prime factors as a hidden but straightforward clue.

Final Answer:
The number whose prime factors include the additional prime 31 and which therefore is the odd man out is 186.