Pick the odd number: Exactly one of the following is a perfect cube (n^3). The others are not perfect cubes. Identify the perfect cube.

Introduction / Context:Perfect powers provide clean classification cues. A perfect cube equals n^3 for some integer n. Among the options, exactly one is a perfect cube; the rest are not. The task is to identify the perfect cube.

Given Data / Assumptions:

• Options: 729, 123, 423, 621.
• Recall of common cubes: 8 = 2^3, 27 = 3^3, 64 = 4^3, 125 = 5^3, 216 = 6^3, 343 = 7^3, 512 = 8^3, 729 = 9^3, 1000 = 10^3.

Concept / Approach:Use recognition of standard cubes or apply prime factorization: a perfect cube’s prime exponents are all multiples of 3. A quick digit-sum or parity check alone is insufficient for cubes, so rely on known landmarks or factorization.

Step-by-Step Solution:

Verification / Alternative check:Prime factorization: 729 = 3^6, which is (3^2)^3, hence a perfect cube. The others do not factor into primes with all exponents multiples of 3.

Why Other Options Are Wrong:123, 423, and 621 are not exact cubes of integers; each sits strictly between consecutive cubes.

Common Pitfalls:Assuming that “divisible by 3” or having a certain digit sum implies being a cube. Divisibility helps but does not prove cube status; exact equality to n^3 is required.

Final Answer:729 is the only perfect cube in the list.