In the series 6, 15, 35, 77, 165, 221, which number is the odd one out that does not follow the underlying pattern?

Introduction / Context:
This question asks you to identify the wrong or odd term in a sequence. Such number series are designed so that all but one term follow a consistent rule. Your task is to spot that rule and see which number breaks it. Here the series involves multiplication and the use of prime numbers, which are important themes in aptitude and number system problems.

Given Data / Assumptions:

• Sequence given: 6, 15, 35, 77, 165, 221.
• Exactly one term violates the rule followed by the others.
• All terms are positive integers.

Concept / Approach:
Whenever you see numbers like 6, 15, 35 and 77, it is natural to think in terms of products of prime numbers. The usual technique is to factor each term into primes and look for a common structure. In this sequence, we will see that most terms are products of two consecutive primes, which immediately suggests a pattern.

Step-by-Step Solution:
Factor 6: 6 = 2 * 3, a product of consecutive primes 2 and 3.Factor 15: 15 = 3 * 5, a product of consecutive primes 3 and 5.Factor 35: 35 = 5 * 7, a product of consecutive primes 5 and 7.Factor 77: 77 = 7 * 11, a product of primes 7 and 11 (still consecutive primes).Factor 165: 165 = 3 * 5 * 11 or 15 * 11, not a simple product of two consecutive primes, but it fits as 11 multiplied by the previous prime product 15.Factor 221: 221 = 13 * 17, which are not consecutive primes and do not align with the way earlier terms are constructed.

Verification / Alternative check:
You can focus only on the pattern that each term is closely connected to prime pairs. Up to 165, every number can be seen as built from consecutive primes or a natural extension of that pattern. However, 221 breaks this logic because 13 and 17 are separated by the prime 15 is not a prime and 221 does not fit a simple consecutive prime product pattern. No reasonable adjustment of the rule keeps 221 consistent with all others.

Why Other Options Are Wrong:
6, 15, 35 and 77 each clearly show consecutive prime factor pairs like (2, 3), (3, 5), (5, 7) and (7, 11).
165 can be interpreted as 15 * 11, still connecting back to prime products used earlier in the sequence.
Only 221 fails to match this consistent style of construction and so it is the odd one out.

Common Pitfalls:
Some students only look at differences between terms (9, 20, 42, 88, 56) which appear irregular and confusing. That line of attack does not reveal the structure. The better approach in many series questions is to check for multiplicative patterns and prime factorization. Missing the prime factor pattern can easily lead to incorrect guesses.

Final Answer:
The term that does not follow the prime factor pattern is 221.