Identify the odd number: Exactly one of 26, 124, 728, and 64 is both a perfect square and a perfect cube (i.e., a perfect sixth power). Choose that distinct number.

Introduction / Context:
Some integers are “perfect powers,” such as perfect squares n^2, perfect cubes n^3, or numbers that are both (n^6). Among these four, one number uniquely satisfies both square and cube properties.

Given Data / Assumptions:

• Candidates: 26, 124, 728, 64.
• 64 = 8^2 and 4^3 (also 2^6).

Concept / Approach:
Check if any number matches known squares or cubes. Being both a square and a cube means it is a perfect sixth power.

Step-by-Step Solution:

Verification / Alternative check:
Prime factorization: 64 = 2^6 has exponents multiples of both 2 and 3, confirming square and cube status simultaneously.

Why Other Options Are Wrong:
They are neither perfect squares nor perfect cubes.

Common Pitfalls:
Assuming proximity to a well-known square or cube (e.g., 728 vs 729) implies equality.

Final Answer:
64 is the number that is both a perfect square and a perfect cube.