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

8th August, 2022

## 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.

## 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 30

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

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