Perfect cube by multiplication: By what least number must 21600 be multiplied to make it a perfect cube?

Introduction / Context: Here we must multiply, not divide, to turn a number into a perfect cube. The approach is to complete prime exponents up to the next multiples of 3.

Given Data / Assumptions: N = 21600.

Concept / Approach: Prime factorize N and raise deficient exponents (mod 3) by multiplying with the smallest factor so each becomes a multiple of 3.

Step-by-Step Solution: 21600 = 216 × 100 = (2^3 × 3^3) × (2^2 × 5^2) = 2^5 × 3^3 × 5^2.To reach multiples of 3: 2^5 needs one more 2 (to 2^6), 3^3 is fine, 5^2 needs one more 5 (to 5^3).Least multiplier = 2 × 5 = 10.

Verification / Alternative check: (2^5 × 3^3 × 5^2) × (2 × 5) = 2^6 × 3^3 × 5^3 = (2^2 × 3 × 5)^3 = (60)^3.

Why Other Options Are Wrong: 6, 15, 30, 60 are not minimal, though some also achieve a cube; 10 is the least.

Common Pitfalls: Forgetting to check all prime bases or overshooting by multiplying with larger factors than necessary.

Final Answer: 10