 # Finding The Perfect Cube of Numbers

A cube is a three dimensional figure which has all the edges of equal length. The volume of a cube is given by the product of its dimensions. For example, for a cube of edge length ‘a’ units the volume of the cube is given by “a × a × a “which is equal to a3 unit3. Similarly we can calculate the perfect cube of various numbers. Illustration 1: Determine the volume of a cube with edge length 7 cm.

Solution: The side of the cube is 7 cm.

Volume = side × side × side = 7 × 7 × 7 = 343 cm3.

The numbers such as 1, 8, 27, 64, etc. are known as perfect cubes or cube numbers.

When a cube is broken or cut, total volume still remains the same. Hence we can observe that to construct a cube of edge length ‘2a’ with cubes of edge length ‘a’ we need to have 8 cubes of edge length a.

Following is the list of perfect cube numbers from 1 to 10.

 Number Cube 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000

## Finding if a number is perfect cube?

Every number can be expressed as the product of power of it’s Prime factors. If the power of all the Prime factors is in the multiple of 3, then the number is said to be a Perfect cube.

1) 15 = 3 × 5

153 = 15 × 15 × 15 = 3 × 5 × 3 × 5 × 3 × 5 = 33 × 53

2) 18 = 2 × 3 × 3

183 = 18 × 18 × 18 = 2 × 3 × 3 × 2 × 3 × 3 × 2 × 3 × 3 = 23 × 33 × 33

From the above examples we can observe that each prime factor of a number appears three times in the prime factorization of its cube.

Illustration 2: Is the number 600 a perfect cube?

Firstly, stating the prime factors of the number 600.

$600 = 2 \times 2\times 2 \times 3 \times 5 \times 5 = 2^{3}.3.5^{2}$

The number 600 is not a perfect cube as all the prime factors is not a multiple of three.

Illustration 3: What should be multiplied to the number 120 to make it a perfect cube?

Factors of $120 = 2 \times 2 \times 2 \times 3 \times 5 = 2^{3}. 3.5$

The factor 2 occurs thrice, 3 and 5 occurs only once. So in order to make it a perfect cube we have to multiply 120 square of 3 and 5.

Therefore, $120 \times 3 \times 3 \times 5 \times 5 = 27000$<

The number 27000 is the required perfect cube.