What is a Cube Number?

A cube number or a perfect cube is a number that can be written as the cube of an integer. Mathematically, if is an integer, then:

Examples of Perfect Cubes:

• 1³ = 1

• 2³ =8

• 3³ = 27

• 4³ =64

• 5³ =125

These numbers are called perfect cubes because they can be represented as the cube of whole numbers.

Understanding Cubes in Geometry:

• A cube is a three-dimensional solid figure that has all its sides equal.

• It has 6 faces, 12 edges, and 8 vertices.

• A cube of side 2 cm consists of 8 smaller cubes of side 1 cm .

• A cube of side 3 cm consists of 27 smaller cubes of side 1 cm .

 Hardy–Ramanujan Number:

• 1729 is known as the smallest Hardy–Ramanujan number.

• This number is special because it can be written as the sum of two cubes in two different ways:

  • 1729 = 1³ +12³

  • 1729 = 9³ +10³

• Other such numbers include 4104, 13832, 20683, etc.

4. Properties of Perfect Cubes:

• Some examples of perfect cubes:

  • 1³ =1

  • 2³ =8

  • 3³ = 27

  • 4³ = 64

  • 5³ =125

  • 6³ =216

  • 7³ =343
  • 8³ =512

  • 9³ =729

  • 10³ =1000

• There are only ten perfect cubes between 1 and 1000.

• Some numbers, like 9, are not perfect cubes because there is no integer whose cube equals 9.

5. Cube Properties of Even and Odd Numbers:

• The cube of an even number is always even.

  • Example: , ,

• The cube of an odd number is always odd.

  • Example: , ,

6. Observations on One’s Digit of Cube Numbers:

• The unit digit of a cube number depends on the unit digit of the original number:

  • If a number ends in 1, its cube ends in 1.

  • If a number ends in 2, its cube ends in 8.

  • If a number ends in 3, its cube ends in 7.

  • If a number ends in 4, its cube ends in 4.

  • If a number ends in 5, its cube ends in 5.

  • If a number ends in 6, its cube ends in 6.

  • If a number ends in 7, its cube ends in 3.

  • If a number ends in 8, its cube ends in 2.

  • If a number ends in 9, its cube ends in 9.

  • If a number ends in 0, its cube ends in 0.

Thus, analyzing the unit digits of cube numbers helps in identifying patterns.

7. Additional Observations:

• Negative numbers can also have perfect cubes:

  • Example: ,( -2)³ = -8 ,  (-3)³ = -27

• The cube root of a perfect cube is always an integer.

• If a number is a perfect cube, each prime factor in its prime factorization appears three times.