Q. What is the least number by which 3600 be divided to make it a perfect cube

To make 3600 a perfect cube, you need to divide it by 450. Here's why: First, find the prime factorization of 3600, which is 2⁴ × 3² × 5². For a number to be a perfect cube, all exponents in its prime factorization must be divisible by 3. Currently, the exponent of 2 is 4 (which gives remainder 1 when divided by 3), the exponent of 3 is 2 (remainder 2), and the exponent of 5 is 2 (remainder 2).

To achieve exponents divisible by 3, you need to reduce each exponent to the nearest multiple of 3 below it: reduce 2⁴ to 2³ (removing one factor of 2), reduce 3² to 3⁰ (removing two factors of 3), and reduce 5² to 5⁰ (removing two factors of 5). The number you must divide out is therefore 2¹ × 3² × 5² = 2 × 9 × 25 = 450. After dividing 3600 by 450, you get 8, which equals 2³, a perfect cube. This problem demonstrates how prime factorization helps identify what operations are needed to transform numbers into perfect powers, a useful technique in algebra and number theory for simplifying expressions and solving equations involving roots and exponents.