A cube is a solid figure which has all its sides equal. In mathematical terms, the cube of a number is the result of multiplying a whole number by itself twice, i.e, the whole number is used three times just like the sides of a cube. For example:

- Cube of 1 = 1
^{3}= 1 * 1 * 1 = 1, - Cube of 2 = 2
^{3}= 2 * 2 * 2 = 8, - Cube of 3 = 3
^{3}= 3 * 3 * 3 = 27……. and so on.

**Properties of T**he C**ube**

- Cubes of all odd numbers are odd.
- Cubes of all even numbers are even.
- Cubes of numbers ending with 2 will end in 8. Similarly, cubes of numbers ending with 8 always end with 2.
- Cubes of numbers ending with 3 will end in 7. Similarly, cubes of numbers ending with 7 always end with 3.

**Unit Digits in Cube Numbers**

As we discussed in the properties of the cube

- If a number is odd, its cube number unit digit is also odd.
- And similarly, If a number is even, its cube number unit digit is also even.

Following is a table that shows the unit digit of a number and the unit digit of the cube of that number.

The units digit of the number | The units digit of its cube |
---|---|

1 | 1 |

2 | 8 |

3 | 7 |

4 | 4 |

5 | 5 |

6 | 6 |

7 | 3 |

8 | 2 |

9 | 9 |

**Perfect Cubes**

A perfect cube is an integer that is equal to some other integer raised to the third power. For example: 125 is a perfect cube since 125 = 5 * 5 * 5 = 5^{3}. However, 121 is not a perfect cube because there is no integer n such that n^{3} = 121. Some other examples of perfect cubes are 1, 8, 27, 64……

**Cube Root**

The cube root of a number is a value that, when multiplied three times, gives that number. For example 3 * 3 * 3 = 27, so the cube root of 27 is 3. The symbol ‘** ^{3}√**‘ denotes ‘cube root‘.

**Finding cube root by prime factorization:**

Prime factorization is a method through which you can easily determine whether a particular figure represents a perfect cube. If each prime factor can be clubbed together in groups of three, then the number is a perfect cube. For example: Let us consider the number 1728. 1728 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 = 2^{3} * 2^{3} * 3^{3}. Here, all the numbers can be clubbed in groups of three. So we can definitely say that 1728 is a perfect cube. In fact, the cube root of 1728 is 12.

**More Examples:**

Example 1:216 = 2 * 2 * 2 * 3 * 3 * 3 = 2

^{3}* 3^{3 }= 6^{3}Hence the cube root of 216 is 3

Example 2:5832 = 2 * 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3 = 2

^{3}* 3^{3}* 3^{3 }= 18^{3}Hence the cube root of 216 is 18

**Cube Root of a Cube Number using estimation:**

- Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number. So 117649 has two groups, and first group(649) and the second group(117).
- The unit’s digit of the first group (649) will decide the unit digit of the cube root. Since the number 649 ends with 9, the cube roots unit’s digit is 9.
- Find the cube of numbers between which the second group lies. The other group is 117. We know that 43=64 and 53=125. 64 < 117 < 125. Take the smaller number between 4 and 5 as the ten’s digit of the cube root. So, 49 is the cube root of 117649.

**Hardy – Ramanujan Numbers**

Numbers like 91, 1729, 4104, 13832 are known as Hardy – Ramanujan Numbers because they can be expressed as sum of two cubes in two different ways. For example,** 91 can be expressed as the sum of two cubes (6, -5) and (4, 3)**.

- 91 = 6
^{3 }+ (-5)^{3 }= 4^{3 }+ 3^{3}- 1729 = 1
^{3 }+ 12^{3 }= 9^{3 }+ 10^{3 }