Key Points

The cube of a number is that number raised to the power of 3. For any number $n$, its cube is calculated as $n \times n \times n$, which is written as $n^3$.

A natural number is called a perfect cube if it is the cube of a natural number. For example, $27$ is a perfect cube because $27 = 3^3$, but $9$ is not a perfect cube.

The cube of an even number is always an even number, for instance, $4^3 = 64$. Similarly, the cube of an odd number is always an odd number, for instance, $5^3 = 125$.

The digit in the one's place of a perfect cube is determined by the one's digit of the original number. For example, a number ending in 2 will have a cube ending in 8 (since $2^3 = 8$), and a number ending in 7 will have a cube ending in 3 (since $7^3 = 343$).

A number is a perfect cube if and only if each of its prime factors appears three times (or a multiple of three times) in its prime factorization. For example, the prime factorization of $216$ is $2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3$, so it is a perfect cube.

To determine if a number is a perfect cube, perform its prime factorization. If all prime factors can be grouped into sets of three (triplets), the number is a perfect cube.

To find the smallest number by which a given number must be multiplied to become a perfect cube, first find its prime factorization. Then, identify the factors that do not form a triplet and multiply by the missing factors to complete the triplets.

To find the smallest number by which a given number must be divided to become a perfect cube, find its prime factorization. The required divisor is the product of the prime factors that are left over after grouping the factors into triplets.

The cube root of a number is the value that, when cubed, gives the original number. It is the inverse operation of cubing. The symbol for the cube root is $\sqrt[3]{}$. For example, $\sqrt[3]{125} = 5$ because $5^3 = 125$.

To find the cube root of a perfect cube using prime factorization, group the factors into triplets. The cube root is the product obtained by taking one factor from each triplet. For example, $\sqrt[3]{3375} = \sqrt[3]{3^3 \times 5^3} = 3 \times 5 = 15$.

The Hardy-Ramanujan numbers are numbers that can be expressed as the sum of two cubes in two different ways. The smallest such number is 1729, which can be written as $1729 = 1^3 + 12^3$ and also as $1729 = 9^3 + 10^3$.

A perfect cube can be expressed as the sum of consecutive odd numbers. For example, $1^3 = 1$, $2^3 = 3+5$, $3^3 = 7+9+11$, and $4^3 = 13+15+17+19$.

A perfect cube cannot end with two zeros. The number of zeros at the end of a perfect cube must be a multiple of three. For instance, $10^3=1000$ (three zeros) is a perfect cube, but $100$ is not.