Pick the odd number: Exactly one of 729, 123, 423, and 621 is a perfect cube n^3; the others are not. Identify the perfect cube.

Introduction / Context:Recognizing perfect powers is a frequent classification skill. Among the four numbers, only one is exactly n^3 for an integer n.

Given Data / Assumptions:

• Options: 729, 123, 423, 621.
• Recall standard cubes: 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Concept / Approach:Compare against known cubes or use factorization; in a perfect cube, all prime exponents are multiples of 3.

Step-by-Step Solution:

Verification / Alternative check:Prime factorization: 729 = 3^6 = (3^2)^3. The others do not yield exponents all multiples of 3.

Why Other Options Are Wrong:They are not exact cubes of integers.

Common Pitfalls:Assuming proximity to a cube implies equality; only exact equality counts.

Final Answer:729 is the perfect cube and hence the odd number.