In the list of numbers 12, 48, 168, 502 and 1260, which one is the odd man out based on divisibility by 3?

Introduction / Context:
This odd man out problem focuses on the basic concept of divisibility by 3. The numbers 12, 48, 168, 502 and 1260 vary in size, but one of them differs from the others with respect to divisibility. Recognising this simple property allows us to quickly identify the outlier.

Given Data / Assumptions:
The numbers under consideration are:

• 12
• 48
• 168
• 502
• 1260

We assume that all but one of these numbers are divisible by 3, and that divisibility by 3 is the central property being tested.

Concept / Approach:
The divisibility rule for 3 states that a number is divisible by 3 if and only if the sum of its digits is a multiple of 3. To solve the problem, we simply apply this rule to each number in turn. The number that fails the test is the odd one out.

Step-by-Step Solution:
Step 1: Check 12. The sum of its digits is 1 + 2 = 3, which is divisible by 3, so 12 is a multiple of 3.Step 2: Check 48. The sum of its digits is 4 + 8 = 12, which is divisible by 3, so 48 is a multiple of 3.Step 3: Check 168. The sum of its digits is 1 + 6 + 8 = 15, which is divisible by 3, so 168 is a multiple of 3.Step 4: Check 1260. The sum of its digits is 1 + 2 + 6 + 0 = 9, which is divisible by 3, so 1260 is a multiple of 3.Step 5: Check 502. The sum of its digits is 5 + 0 + 2 = 7, which is not divisible by 3, so 502 is not a multiple of 3.

Verification / Alternative check:
We can also verify by direct division. 12 / 3 = 4, 48 / 3 = 16, 168 / 3 = 56 and 1260 / 3 = 420, all of which are integers. In contrast, 502 / 3 gives a non integer result. These calculations confirm that 502 is the only number in the list that fails to be divisible by 3.

Why Other Options Are Wrong:
Numbers 48, 168 and 1260 all pass the divisibility by 3 test, just like 12. They therefore share a common property and belong to the same group. Removing any of them as the odd man out would leave 502, which still has a fundamentally different divisibility property compared with the rest.

Common Pitfalls:
Sometimes students overcomplicate such questions by looking for sophisticated series patterns or factor structures. It is important to first test very basic properties like divisibility by 2, 3, 5, 9 or 11. In this example, a simple application of the digit sum rule immediately reveals the correct answer.

Final Answer:
The only number that is not divisible by 3 is 502.