Classification – Odd number out (perfect powers vs. prime): Which number does not belong: 121, 324, 523, 729?

Introduction / Context:
In numeric classification, a frequent separator is “perfect power” versus “non-power/prime.” Three of these numbers are perfect powers (squares or higher powers), while one is a prime number that is not a perfect power. Identifying that prime isolates the outlier.

Given Data / Assumptions:

• Numbers: 121, 324, 523, 729.
• Perfect powers to test: squares (n^2), cubes (n^3), and common higher powers.

Concept / Approach:
Check each number: determine if it is a perfect square or cube. If it is neither, assess primality. The one that is prime (and not a perfect power) is the odd one out.

Step-by-Step Solution:
121 = 11^2 → perfect square.324 = 18^2 → perfect square.729 = 27^3 (also 3^6) → perfect power.523 → not a square (22^2 = 484, 23^2 = 529), not a simple cube; test divisibility: not divisible by 2, 3 (sum of digits 10), 5 (ends not 0/5), 7, 11, 13, 17, 19, 23. 523 is prime.

Verification / Alternative check:
Because 523 is prime, it cannot be expressed as a^b with integers a ≥ 2, b ≥ 2. The other three are unambiguously perfect powers.

Why Other Options Are Wrong:

• 121 and 324 are perfect squares; 729 is a perfect cube (and higher power). All share the perfect-power property.

Common Pitfalls:
Mistaking 729 as a square because it lies near 27^2 = 729 (false; 27^2 is 729? No, 27^2 = 729 is true, which still makes it a perfect power; in fact 729 = 3^6 and 27^2). Regardless, it is a perfect power.

Final Answer:
523