To find the smallest value of K such that 20K is a perfect cube, we need to determine the prime factors of 20K and then adjust K accordingly.
The prime factorization of 20 is 2^2 * 5, and to make it a perfect cube, we need to have all the prime factors raised to multiples of 3.
For the factor of 2, we need it raised to the power of 3, so K = 2^(3-2) = 2.
For the factor of 5, we need it raised to the power of 3, so K = 5^(3-1) = 5^2 = 25.
So, the smallest value of K such that 20K is a perfect cube is K = 2 * 25 = 50.