# Lowest Common Multiple (LCM) and Highest Common Factor (HCF)

20th April, 2021

## How do you find the Lowest Common Multiple (LCM) or Highest Common Factor (HCF)?

Before showing what Lowest Common Multiple (LCM) and Highest Common Factor (HCF) are, I will explain the difference between factors and multiples.

I will then talk about prime numbers and prime factors.

Table of Contents

## What are factors?

The **factors** of a number are those that divide into another.

e.g. the factors of 12 are 1, 2, 3, 4, 6 and 12

## Pairs of factors

Factors are often written as “pairs”, i.e. two numbers multiplied to make that number.

e.g. the factors of 12, listed as pairs, are:

1 **and** 12

2 **and** 6

3 **and** 4

These are the same numbers we listed above, but in numerical order.

## How to find pairs of factors of a number

Use this methodical approach, listing them in pairs, and start with 1 and the number.

e.g. “Find the factors of 60”

The first pair is **1** and **60**

Next ask, “Is it divisible by 2?” It is even, so yes, it is divisible by 2. Divide 60 by 2 which is 30. The next pair is **2** and **3o**

“Is it divisible by 3?” Yes, it is divisible by 3 (using the trick of adding the digits together). The next pair is **3** and **20**

“Is it divisible by 4?” When we divided it by 2 we got 30, an even number, so yes, we can divide it again so it must be divisible by 4. The next pair is **4** and **15** (half of 30)

“Is it divisible by 5?” It ends with a 0, so it is divisible by 5. The next pair is **5** and **12**

“Is it divisible by 6?” Yes, it is divisible by 6 (using the trick of adding the digits together and it is even). The next pair is **6** and **10**

“Is it divisible by 7?” No, 56 is in the 7 times table, and 60 is only 4 more.

“Is it divisible by 8?” No, 64 is in the 8 times table, and 60 is only 4 less.

“Is it divisible by 9?” No, it is not divisible by 9 (using the trick of adding the digits together).

( Working methodically this way, notice how the pairs of numbers get closer together. Once you reach 9 there is no need to continue as we already have 10 and we have reached the bottom of our list of pairs – see diagram. )

**factors of 60**

1 and 60

2 and 3o

3 and 20

4 and 15

5 and 12

6 and 10

## What is the Highest Common Factor (HCF) of two numbers?

The **highest common factor** (HCF) is the BIGGEST number that will DIVIDE INTO ALL numbers in the question.

e.g. Factors of 36 are: 1, 2, 3, 4, 6, 9, 12, **18**, 36

and Factors of 54 are: 1, 2, 3, 6, 9, **18**, 27, 54

therefore the Highest Common Factor of 36 and 54 is **18**.

## Tip for remembering what the Highest Common Factor (HCF) means

Read Highest Common Factor (**HCF**) backwards

‘**factor**’ – list factors of both numbers

‘**common**’ – factor appears in both lists

‘**highest**’ – the highest of those common factors

## What are multiples?

The **multiples** of a number are numbers that can be made from multiplying 1 or more of that number (essentially the times tables of that number)

e.g. the multiples of 3 are

3, 6, 9, 12, …. 30, 33, 36, …. 60, …. 99, … 333 etc!

## How to list multiples

Start with the number – that is the ** first** multiple. And then repeatedly add the number to the previous sum.

e.g. the multiples of 17 are:

17, 34, 51, 68, 85, 102, …

(quick tip for repeatedly adding a number like 17, which isn’t that easy in your head – add 20 and then take away 3)

## What is the Lowest Common Multiple (LCM) of two numbers?

The **lowest common multiple** (LCM) is the SMALLEST number that will DIVIDE BY ALL the numbers in the question.

e.g. Multiples of 6 are : 6, 12, 18, 24, 30, 36, **42**, 48, 54, 60, 66, 72, 78, 84, …

and Multiples of 7 are : 7, 14, 21, 28, 35, **42**, 49, 56, 63, 70, 77, 84, …

therefore the Lowest Common Multiple of 6 and 7 is **42**.

## Tip for remembering what the Lowest Common Multiple (LCM) means

Read Lowest Common Multiple backwards

‘**multiple**’ – list a few multiples of both numbers

‘**common**’ – multiple appears in both lists

‘**lowest**’ – the smallest number

## What’s the difference between factors and multiplies?

To sum up: **factors** divide into another (remember “pairs of factors” multiply together) and **multiples** are when you multiply the number by 1, by 2, by 3 etc – e.g. the times tables of that number.

## What is a prime number?

A **prime number** is any number that can only be divided by 1 and itself – it has **exactly two factors**.

e.g. these are the first few prime numbers (it is an infinite sequence!)

2 3 5 7 11 13 17 19 23 29 31 37 41…

## Are all prime numbers odd? Are all odd numbers prime?

2 is the only even prime number. All other prime numbers are odd.

But, not all odd numbers are prime,

e,g, 9 is odd but not prime, as it is divisible by 1,3 and 9

## Is 1 a prime number?

No, **1 is NOT a prime number**, as it is *only* divisible by itself. It must have one and only one pair of factors.

## What is a prime factor?

A **prime factor** is any factor of a number that is also a prime number.

e.g. the factors of 30 are

1 and 30

**2** and 15

**3** and 10

**5** and 6

The only numbers in this list that are prime are **2**, **3** and **5** – these are the prime factors of 30

## FREE Resource on How to Find Lowest Common Multiple (LCM), Highest Common Factor (HCF), and Prime Factors

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