The smallest number by which 3600 be divided to make it a perfect cube—

Concept: A perfect cube is a number whose prime factorization has every exponent as a multiple of 3. To find the smallest number by which N must be divided so the quotient becomes a perfect cube, write N in prime-factorized form and, for each prime, remove just enough of its exponent to bring it down to the nearest lower multiple of 3 (an exponent already a multiple of 3 needs nothing removed).

1. Prime-factorize 3600: 3600 = 36 × 100 = (22 × 32) × (22 × 52) = 24 × 32 × 52.

2. Prime 2 has exponent 4; the nearest lower multiple of 3 is 3, so one factor of 2 (21) must be removed.

3. Prime 3 has exponent 2; the nearest lower multiple of 3 is 0, so both factors of 3 (32) must be removed.

4. Prime 5 has exponent 2; the nearest lower multiple of 3 is 0, so both factors of 5 (52) must be removed.

5. The smallest divisor is the product of the removed powers: 2 × 32 × 52 = 2 × 9 × 25 = 450.

Cross-check: 3600 ÷ 450 = 8 = 23, and every exponent in 23 is a multiple of 3, confirming 8 is a perfect cube. Removing any smaller amount would leave at least one exponent not divisible by 3, so 450 is the least such divisor.

Result: 450 is the smallest number by which 3600 must be divided to make it a perfect cube.