Find the odd number: Three of the numbers can be expressed as the product of three consecutive positive integers (k * (k+1) * (k+2)); one cannot. Identify the number that does not fit this pattern.

Introduction / Context:
Structural-factor patterns are common in classification problems. Here, three options equal k * (k+1) * (k+2) for some positive integer k (i.e., product of three consecutive integers). One option does not match this structure. We must identify the mismatch.

Given Data / Assumptions:

• Numbers: 24, 60, 124, 210.
• We search for integers k such that k * (k+1) * (k+2) equals the target.

Concept / Approach:
Test nearby triplets efficiently: for a given target N, find a central cube root to anchor k and check products around that value. Alternatively, check known small products of three consecutive integers.

Step-by-Step Solution:

Verification / Alternative check:
Observe that products of three consecutive integers are always divisible by 3 and by 2 (hence even and a multiple of 3). While 124 is even, it is not divisible by 3 (1 + 2 + 4 = 7), which already suggests it cannot be k * (k+1) * (k+2).

Why Other Options Are Wrong:
They are correct examples of the pattern and thus not the odd item.

Common Pitfalls:
Checking only proximity (e.g., 124 being close to 120) without verifying exact factorization into consecutive integers.

Final Answer:
124 does not equal a product of three consecutive integers and is the odd number.