# LCM and HCF

**The lowest common multiple**

The **LCM** **(****lowest common multiple)** of two numbers is the lowest number that is a multiple of both the numbers.

For small numbers, this can be found by writing down the first few multiples for both numbers until we find a number that is in both lists.

For example,

Multiples of 20 are : 20, 40, 60, 80, 100, 120 . . .

Multiples of 25 are : 25, 50, 75, 100, 125 . . .

The LCM of 20 and 25 is **100**.

If there are two numbers have no common factors (except 1) then the lowest common multiple of the two numbers will be the product of the two numbers.

Therefore, 4 and 5 have no common factors and so the lowest common multiple of 4 and 5 is 4 × 5, 20.

We use the **lowest common multiple** when adding and subtracting fractions.

For example,

The Lowest common multiple of 9 and 12 is 36.

Points to note:

- To add two fractions together they must have the same denominator
- The top and the bottom must be multiplied by the same number

The highest common factor

The **HCF (****highest common factor)** of two numbers is the highest number that is a factor of both numbers.

The highest common factor can be found from two numbers by writing down all their factors and finding the largest factor in both lists.

For example,

The Highest common factor of 36 and 45 is **9**.

Note that 3 is also a common factor of 36 and 45 but 9 is the HCF

The highest common factor is used when cancelling fractions.

For example,

The HCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12.

**Using prime factors to find the HCF and LCM**

To find the HCF and LCM of larger numbers we can use the prime factor decomposition.

For example,

Find the HCF and the LCM of 60 and 125.

2 | 60 |

2 | 30 |

3 | 15 |

5 | 5 |

1 |

60 = 2 x 2 x 3 x 5

2 | 294 |

3 | 147 |

7 | 49 |

7 | 7 |

1 |

294 = 2 x 3 x 7 x 7

**The LCM of co-prime numbers**

**Co-prime** or **relatively prime** numbers are two numbers which have a highest common factor (or HCF) of 1.

For two whole numbers *a* and *b* we can write:

If two whole numbers *a* and *b* are co-prime then:

For example, the numbers 8 and 9 do not share any common multiples other than 1. They are co-prime.

Therefore, LCM(8, 9) = 8 × 9 = 72

Points to note:

- If two numbers are co-prime, that is they do not share any common factors other than 1, their LCM is equal to the product of the two numbers.
- This rule is also true if more than two numbers are co-prime.

**The LCM of numbers that are not co-prime**

If two numbers are not **co-prime** then their highest common factor is greater than 1.

If two numbers *a* and *b* are *not* co-prime then their lowest common multiple is equal to the product of the two numbers divided by their highest common factor.

This can be written as:

Points to note:

This formula can be re-arranged by saying that the LCM(*a,b*) × HCM(*a,b*) = *ab* (for any two whole numbers *a* and *b*, the LCM of *a* and *b* multiplied by the HCF of *a* and *b* equals the product of *a* and *b*.)