Among the numbers 396, 462, 572, 427, 671 and 264, which number is the odd one out based on a common divisibility rule?

Introduction / Context:
This odd man out question revolves around a divisibility property rather than simple size or parity. Many such questions use divisibility by a particular prime or composite number as the hidden pattern. Here, examining divisibility by 11 reveals a clear difference between most of the numbers and one special case.

Given Data / Assumptions:

• The numbers are 396, 462, 572, 427, 671, and 264.
• We are asked to find the single number that does not share a common property with the others.
• All numbers are positive integers.
• We suspect a divisibility rule may be the basis for the pattern.

Concept / Approach:
A useful starting point is to test for divisibility by a moderately sized number that might be a common factor, such as 3, 7, 9, or 11. For 11 in particular, there is a simple rule: for a number with digits d1, d2, d3, ..., the alternating sum of digits determines divisibility. If the absolute value of (sum of digits in odd positions - sum of digits in even positions) is a multiple of 11, then the number is divisible by 11. Applying this test or direct division to each number can show which numbers share a common factor of 11.

Step-by-Step Solution:
Check 396 for divisibility by 11. We can also directly divide: 396 / 11 = 36, so 396 is divisible by 11. Check 462: 462 / 11 = 42, so 462 is divisible by 11. Check 572: 572 / 11 = 52 exactly, so 572 is divisible by 11. Check 671: 671 / 11 = 61 exactly, so 671 is divisible by 11. Check 264: 264 / 11 = 24, so 264 is also divisible by 11. Now check 427: 427 / 11 is not an integer and leaves a remainder. Thus 427 is not divisible by 11. Therefore, all the numbers except 427 are divisible by 11, making 427 the odd man out.

Verification / Alternative Check:
We can use the standard divisibility rule for 11 as an alternative check. For 396, compute (3 + 6) - 9 = 0, which is a multiple of 11, so it is divisible. For 462, compute (4 + 2) - 6 = 0, again divisible. For 572, (5 + 2) - 7 = 0. For 671, (6 + 1) - 7 = 0. For 264, (2 + 4) - 6 = 0. But for 427, (4 + 7) - 2 = 9, which is not a multiple of 11, confirming that 427 is not divisible by 11. This reinforces our conclusion that 427 is the only number that does not share divisibility by 11.

Why Other Options Are Wrong:

• Option 671: This number is divisible by 11 (671 = 11 * 61), so it matches the common property and cannot be the odd one out.
• Option 462: This is 11 * 42 and clearly shares the same divisibility property.
• Option 264: This equals 11 * 24, again fitting the pattern.
• Option 396: This equals 11 * 36 and is also part of the majority group.

Common Pitfalls:
Some students may try to look for patterns in the differences between numbers or for other numerical tricks, which can obscure the simpler common factor property. Others may not remember the divisibility rule for 11 and overlook this clue. It is a good practice to check divisibility by small primes and by 11 in such questions. Also, relying solely on mental division without verification can lead to mistaken judgments about which numbers are divisible.

Final Answer:
The odd man out based on divisibility by 11 is 427.