#### description

- Adding consecutive odd numbers
- Cubes and their prime factors

#### notes

**i) Adding consecutive odd numbers: **

`1 = 1 = 1^3`

`3 + 5 = 8 = 2^3`

`7 + 9 + 11 = 27 = 3^3`

`13 + 15 + 17 + 19 = 64 = 4^3`

`21 + 23 + 25 + 27 + 29 = 125 = 5^3`

**ii) Cubes and their prime factors :**

Consider the following prime factorisation of the numbers and their cubes.

Prime factorisation of a number | Prime factorisation of its cube |

4 = 2 × 2 | `4^3 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^3 × 2^3` |

6 = 2 × 3 | `6^3 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2^3 × 3^3` |

15 = 3 × 5 | `15^3 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3^3 × 5^3` |

12 = 2 × 2 × 3 | `12^3 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2^3 × 2^3 × 3^3 ` |

Do you remember that `a^m × b^m = (a × b)^m` .

Each prime factor of a number appears three times in the prime factorisation of its cube.