To find the smallest number by which a given number should be divided so that the quotient is a perfect cube, we need to understand the concept of perfect cubes and prime factorization.

1. Perfect Cubes:
A perfect cube is a number that can be expressed as the cube of an integer. For example, 8 is a perfect cube because it can be expressed as 2^3 (2 multiplied by itself three times).

2. Prime Factorization:
Prime factorization is the process of breaking down a number into its prime factors. Prime factors are the prime numbers that multiply together to give the original number. For example, the prime factorization of 24 is 2^3 * 3 (2 multiplied by itself three times, and then multiplied by 3).

Now, let's solve the given examples:

1. Number: 2000
To find the smallest number by which 2000 should be divided to get a perfect cube, we need to perform prime factorization.

2000 = 2^4 * 5^3

To make the quotient a perfect cube, we need to divide 2000 by the highest power of each prime factor present. In this case, the highest power of 2 is 4, and the highest power of 5 is 3. Therefore, we need to divide 2000 by 2^4 * 5^3.

Smallest number = 2^4 * 5^3 = 2 * 2 * 2 * 2 * 5 * 5 * 5 = 800

So, 2000 should be divided by 800 to get a perfect cube.

2. Number: 5184
Following the same steps, we perform prime factorization for 5184.

5184 = 2^6 * 3^4

The highest power of 2 is 6, and the highest power of 3 is 4. To make the quotient a perfect cube, we need to divide 5184 by 2^6 * 3^4.

Smallest number = 2^6 * 3^4 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 3 = 1728

So, 5184 should be divided by 1728 to get a perfect cube.

By following the steps of prime factorization and dividing by the highest powers of the prime factors, we can find the smallest number by which a given number should be divided to obtain a perfect cube.