In the series 6, 15, 35, 63, 143, 221, 323, only one number is wrong. Identify the wrong number.

Introduction / Context:
This odd man out question is based on a series where most terms are products of consecutive prime numbers. Recognising this prime factorisation pattern allows us to identify the term that does not fit.

Given Data / Assumptions:

• Series: 6, 15, 35, 63, 143, 221, 323.
• Exactly one number is incorrect or does not follow the underlying rule.
• The rule likely involves prime numbers and multiplication.

Concept / Approach:
Since the numbers are not in a simple arithmetic or geometric progression, we consider factorisation. A common competitive exam pattern is a sequence of products of consecutive primes such as 2 * 3, 3 * 5, 5 * 7, and so on. We can test each term against this idea.

Step-by-Step Solution:
Step 1: Factorise 6: 6 = 2 * 3, which are consecutive primes. Step 2: Factorise 15: 15 = 3 * 5, again consecutive primes. Step 3: Factorise 35: 35 = 5 * 7, consecutive primes. Step 4: Factorise 63: 63 = 7 * 9, but 9 is not a prime, so this term breaks the pattern of consecutive prime products. Step 5: Factorise 143: 143 = 11 * 13, consecutive primes. Step 6: Factorise 221: 221 = 13 * 17, consecutive primes. Step 7: Factorise 323: 323 = 17 * 19, consecutive primes. Step 8: The correct term in place of 63 should have been 7 * 11 = 77 to continue the chain 2*3, 3*5, 5*7, 7*11, 11*13, 13*17, 17*19.

Verification / Alternative check:
If we replace 63 with 77, the full corrected series becomes 6, 15, 35, 77, 143, 221, 323. Each term now is the product of two consecutive prime numbers: 2*3, 3*5, 5*7, 7*11, 11*13, 13*17, 17*19. No other single change yields such a clean and consistent pattern, so the identification is clear.

Why Other Options Are Wrong:
The terms 35, 143, and 323 all factorise into products of consecutive primes: 35 = 5 * 7, 143 = 11 * 13, 323 = 17 * 19. Removing any of these would spoil the consecutive prime product chain. Only 63 fails this test because its factorisation 7 * 9 involves a non prime factor.

Common Pitfalls:
Students may initially search for arithmetic or geometric patterns in the raw numbers and overlook factorisation. Another common mistake is to factor only some terms and stop early without noticing the full sequence of prime pairs. In series involving moderate sized composite numbers, checking for prime factor patterns is often a powerful strategy.

Final Answer:
The only number that does not follow the product of consecutive primes pattern is 63.