## Introduction

### Definition

- LCM or LCF is the smallest multiple that two or more numbers divide into evenly i.e. without a remainder.
- A multiple of a number is the product of the original number with another number.
- Some multiples of 4 are 4, 8, 1 2, 1 6, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56 …
- Some multiples of 7 are 7, 1 4, 21 , 28, 35, 42, 49, 56 …
- A Common Multiple is a number that is divisible by two or more numbers. Some common multiples of 4 and 7 are 28, 56, 84, and 1 1 2.
- When looking for the Least Common Multiple, you are looking for the smallest multiple that they both divide into evenly. The least common multiple of 4 and 7 is 28.

**Example**

Find the L.C.M of 8, 12, 18 and 20. (using tables)

**Solution**

2 | 8 | 12 | 18 | 20 |

2 | 4 | 6 | 9 | 10 |

2 | 2 | 3 | 9 | 5 |

3 | 1 | 3 | 9 | 5 |

3 | 1 | 1 | 3 | 5 |

5 | 1 | 1 | 1 | 5 |

1 | 1 | 1 | 1 | 1 |

- The L.C.M is the product of all the divisions used.
- Therefore, L.C.M. of 8, 12, 18 and 20 = 2 x 2 x 2 x 3 x 3 x5

= 2^{3}x 3^{2}x 5

= 360

**Note;**

- Unlike the G.C.D tables, if the divisor /factor does not divide a number exactly, then the number is retained, e.g., 2 does not divide 9 exactly, therefore 9 is retained .
- The last row must have all values 1 .

## Past KCSE Questions on the Topic

- Find the L.C.M of x
^{2}+ x, x^{2}– 1 and x^{2}– x . - Find the least number of sweets that can be packed into polythene bags which contain either 9 or 1 5 or 20 or 24 sweets with none left over.
- A number n is such that when it is divided by 27, 30, or 45, the remainder is always 3. Find the smallest value of n.
- A piece of land is to be divided into 20 acres or 24 acres or 28 acres for farming and Leave 7 acres for grazing. Determine the smallest size of such land.
- When a certain number x is divided by 30, 45 or 54, there is always a remainder of 21 . Find the least value of the number x .
- A number
**m**is such that when it is divided by 30, 36, and 45, the remainder is always 7. Find the smallest possible value of**m**

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