Perfect cube by division: What is the smallest number by which 3600 should be divided so that the result is a perfect cube?

Introduction / Context: Similar to converting to a perfect cube by division, this question focuses on 3600 and finding the minimal divisor to make all prime exponents multiples of 3.

Given Data / Assumptions: N = 3600.

Concept / Approach: Prime factorize N and reduce exponents to the nearest lower multiples of 3 by dividing out the smallest possible factor.

Step-by-Step Solution: 3600 = 36 × 100 = (2^2 × 3^2) × (2^2 × 5^2) = 2^4 × 3^2 × 5^2.To get exponents multiples of 3: reduce 2^4 to 2^3 (divide by 2^1), 3^2 to 3^0 (divide by 3^2), 5^2 to 5^0 (divide by 5^2).Required divisor = 2 × 3^2 × 5^2 = 2 × 9 × 25 = 450.

Verification / Alternative check: 3600 ÷ 450 = 8 = 2^3, a perfect cube.

Why Other Options Are Wrong: 9, 25, 30, 50 do not reduce all exponents to multiples of 3 simultaneously.

Common Pitfalls: Trying to make a perfect square instead of a cube, or omitting one of the prime reductions.

Final Answer: 450