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Lowest common multiple and highest common factor

Tutor: Bilal

# Lowest common multiple and highest common factor

## In a nutshell

The lowest common multiple and highest common factor of two numbers is an application of prime factorisation and Venn diagrams.

## Lowest common multiple (LCM)

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

### Finding the lowest common multiple of two numbers

#### ​​procedure

 1 Find the prime factorisation of both numbers. 2 Draw a Venn diagram, with each circle representing a number. 3 If a prime factor is in both numbers, put the prime factor in the middle part of the Venn diagram (where both circles intersect) once (not twice). If a prime factor is in only one of the numbers, put that prime factor in the correct circle. 4 The lowest common multiple is found by multiplying all the prime factors together.

##### Example 1

What is the lowest common multiple of $16$ and $18$?

First, find the prime factorisation of both numbers using the factor tree method:

$16=2\times2\times2\times2$​​

$18=2\times3\times3$​​

Draw a Venn diagram and put any common prime factors in the middle:

$2$is in both numbers only once, so write down $2$ in the middle of the diagram.

For the $16$ circle, write down the remaining $2$s (three of them).

For the $18$ circle, write down the two $3$s.

Multiply all the numbers together to obtain the LCM:

$2\times2\times2\times2\times3\times3=144$​​

The lowest common multiple of $\underline{16}$ and $\underline{18}$ is $\underline{144.}$

## Highest common factor (HCF)

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

### Finding the highest common factor of two numbers

#### procedure

 1 Find the prime factorisation of both numbers 2 Draw a Venn diagram, with each circle representing a number 3 If a prime factor is in both numbers, put the prime factor in the middle part of the Venn diagram (where both circles intersect) once (not twice). If a prime factor is in only one of the numbers, put that prime factor in the correct circle. 4 The highest common factor is found by multiplying all the prime factors that are only in the middle part of the Venn diagram together.

##### Example 2

What is the highest common factor of $16$ and $18$?

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

The HCF is found by multiplying all the numbers in the middle - which is just $2$.

The highest common factor of $\underline{16}$ and $\underline{18}$ is $\underline{2.}$

The HCF and LCM of two numbers, $a$​ and $b$, follow the formula:

$HCF\times LCM=a\times b$

##### Example 3

The HCF and LCM of $16$ and $18$ are $2$ and $144$ respectively.

$HCF\times LCM=2\times144=\underline{288}$​​

$16\times18=\underline{288}$​​

Hence, the formula works in this case.

Note: You don't need to know this formula, but it may be helpful to use to check if your answer is correct.

## FAQs - Frequently Asked Questions

### What is the lowest common multiple?

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