Smallest divisor for perfect cube

Prime factorization of $2400 = 2^5 \times 3^1 \times 5^2$.

To make it a perfect cube, the powers of each prime factor must be multiples of 3. So we need to divide 2400 by $2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 = 300$.