Difficulty: Medium
Correct Answer: 64
Explanation:
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:
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:
64 → 8^2 and 4^3 → both a square and a cube.26 → not a square (5^2 = 25, 6^2 = 36) and not a cube (2^3 = 8, 3^3 = 27).124 → not a square (11^2 = 121, 12^2 = 144) and not a cube (4^3 = 64, 5^3 = 125).728 → not a square (26^2 = 676, 27^2 = 729) and not a cube (8^3 = 512, 9^3 = 729).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.
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