# LCM and HCF

## In a nutshell

The lowest common multiple (LCM) and highest common factor (HCF) of two (or more) numbers are found by using prime factorisation and Venn diagrams.

## Lowest common multiple (LCM)

### Definition

The lowest common multiple of two (or more) numbers is the smallest number that is divisible by each of them; in other words, it is a multiple of each number.

### Finding the LCM of two numbers: Method 1

#### PROCEDURE

1. | Find the prime factorisation of each number. |

2. | Draw a Venn diagram, with each circle representing a number. |

3. | If a prime factor is in more than one number, place it in the middle part of the Venn diagram (where the circles intersect). If a prime factor is only in one of the numbers, place it in the circle belonging to that number. |

4. | The lowest common multiple is found by multiplying every number inside the Venn diagram. |

##### Example 1

*Find the LCM of the numbers $16$ and $18$.*

*Find the prime factors of each number using the factor tree method.*

*$18=2\times3\times3$*

*$16=2\times2\times2\times2$*

*Draw a Venn diagram like so:*

*Multiply all the numbers together to obtain the LCM.*

*$2\times2\times2\times2\times3\times3=\underline{144}$*

### Method 2

If the numbers you have are quite large, it might be easier to instead write out the prime factors of each number to powers. Then, take each of the factors' greatest powers and multiply them together.

##### Example 2

*Find the LCM of the numbers *$18 \space000$* and *$6048$*.*

*Write out the prime factors of each using prime factor trees.*

*$18\space000 = 2\times 2\times 2\times 2 \times 3\times 3\times 5\times 5\times 5$*

*$6048 = 2\times 2\times 2\times 2\times 2 \times 3 \times 3 \times 3 \times 7$*

*Write them using powers.*

*$18\space000 = 2^4\times 3^2\times 5^3$*

*$6048 = 2^5 \times 3 ^3 \times 7$*

*Multiply the highest power of each prime.*

*$2^5\times3^3\times5^3\times7=\underline{756\space 000}$*

### Finding the LCM of more than two numbers

Both of the above methods can be used to find the LCM of more than two numbers. When using Venn diagrams, simply draw as many circles as there are numbers and ensure that prime factors are placed in the correct overlap.

## Highest common factor (HCF)

### Definition

The highest common factor of two (or more) numbers is the greatest number that divides each of them; or the largest factor.

### Finding the HCF of two numbers - Method 1

#### Procedure

1. | Find the prime factorisation of each number. |

2. | Draw a Venn diagram, with each circle representing a number. |

3. | If a prime factor is in both numbers, place in the middle part of the Venn diagram (where both circles intersect). If a prime factor is in only one of the numbers, place it in the circle belonging to that number. |

3. | The highest common factor is the product of each factor found in the middle part of the Venn diagram. |

##### Example 3

*What is the HCF of $16$ and $18$?*

*The prime factorisation and Venn diagram has already been done, as in the above example.*

*The HCF is found by multiplying all the numbers in the middle - in this case it's simply $\underline{2}$.*

### Method 2

Similarly to finding the LCM, If the numbers you have are quite large, it might be easier to instead write out the prime factors of each number to powers. Then, take each of the repeated factors' smallest powers and multiply them together.

##### Example 4

*Find the HCF of *$18 \space000$ *and* $6048$.

*As above, we have already written out each using powers of prime factors.*

*$18\space000 = 2^4\times 3^2\times 5^3$*

*$6048 = 2^5 \times 3 ^3 \times 7$*

*Multiply the smallest powers of only the repeated prime factors in each number.*

*$2^4\times3^2=\underline{144}$*

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### Finding the HCF of more than two numbers

Again, both of the above methods can be used to find the HCF of more than two numbers. Just remember to draw as many circles as there are numbers and ensure that prime factors are placed in the correct overlap when using Venn.

*Note:** some numbers may have no common factors. In this case, they are known as coprime.*

## Relationship between HCF and LCM

There is a helpful relationship that can be used to check that you have the correct answer:

$LCM\times HCF = a\times b$

Where LCM and HCF are the lowest common multiple and highest common factor of two numbers, $a$ and $b$.

**Tip**: Another way to check you answer is that the HCF is never lower than the LCM!