Year 6 · HCF, LCM and Prime Numbers · Lesson 3.1

Year 6: Common factors and the highest common factor

Part of the Year 6 hcf, lcm and prime numbers chapter. Chapter overview · whole chapter on one page

3.1

Common factors and the highest common factor

Year 6
Identify common factors of two numbers, Know the highest common factor (HCF), Use factors to solve problems

A factor of a number divides into it exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12. They come in pairs that multiply to make 12: 1 × 12, 2 × 6 and 3 × 4.

A common factor is a factor of both numbers. The factors of 12 and 18 are shown in the Venn diagram below. The numbers in the middle are in both lists: 1, 2, 3 and 6. They are the common factors of 12 and 18.

Factors of 12 and 18. The middle holds their common factors. The highest of them is 6, so the HCF is 6.

Counters help us see factor pairs. Twelve counters can make 3 rows of 4, so 3 and 4 are a factor pair of 12. They can also make 2 rows of 6. The number of rows is always a factor, because it divides the counters exactly.

12 counters in 3 rows of 4. Factor pair: 3 and 4.

The highest common factor (HCF) is the biggest number that divides into both numbers exactly. For 12 and 18 the HCF is 6. The HCF tells us the biggest size of equal groups we can share two numbers into.

Key ideas

A factor divides into a number exactly. Write factors in pairs, so that you do not miss any.
A common factor is a factor of both numbers. It is in both lists of factors.
The highest common factor (HCF) is the biggest common factor. 1 is always a common factor.
If the only common factor of two numbers is 1, then their HCF is 1.

Worked example 3.1.1

List the common factors of 12 and 18.

Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.

Method. List the factors of each number, then find the ones that appear in both lists.

List the factors of 12. Use factor pairs so that you do not miss any.
1, 2, 3, 4, 6, 12
List the factors of 18. Use the same method.
1, 2, 3, 6, 9, 18
Find the numbers that are in both lists. These are the common factors.
1, 2, 3, 6

Check. Each common factor divides into both 12 and 18 exactly. ✓

Watch out. 1 is always a common factor, so every list of common factors starts with 1.

Worked example 3.1.2

Find the highest common factor (HCF) of 24 and 36.

Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.

Method. List the factors of each number, find the common factors, then choose the biggest.

List the factors of 24. Use factor pairs so that you do not miss any.
1, 2, 3, 4, 6, 8, 12, 24
List the factors of 36. Use the same method.
1, 2, 3, 4, 6, 9, 12, 18, 36
Find the common factors. These are the numbers that are in both lists.
1, 2, 3, 4, 6, 12
Choose the highest common factor. The HCF is the biggest number in the common list.
HCF = 12

Check. 12 divides into 24 (24 ÷ 12 = 2) and into 36 (36 ÷ 12 = 3). ✓

Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.

Worked example 3.1.3

Find the HCF of 20 and 27.

Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.

Method. List the factors of each number, find the common factors, then choose the biggest.

List the factors of 20. Use factor pairs so that you do not miss any.
1, 2, 4, 5, 10, 20
List the factors of 27. Use the same method.
1, 3, 9, 27
Find the common factors. These are the numbers that are in both lists.
1
Choose the highest common factor. The HCF is the biggest number in the common list.
HCF = 1

Check. 1 divides into 20 (20 ÷ 1 = 20) and into 27 (27 ÷ 1 = 27). ✓

Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.

Watch out

Do not confuse the highest common factor with the lowest. Every list of common factors starts with 1, but the HCF is usually bigger. A common factor must divide into both numbers, not just one of them.

Exercise 3A Factors, common factors and the HCF

Non-calculator

FWrite all the factors of 30.

Done

FWrite all the factors of each number.

(a)16

(b)20

Done

FList the common factors of each pair of numbers.

(a)12 and 16

(b)15 and 25

Done

FFind the HCF of each pair of numbers.

(a)8 and 12

(b)9 and 15

(d)14 and 35

(e)18 and 27

Done

FFind the HCF of 45 and 60. Show your working.

Done

FThe Venn diagram shows the factors of 12 and 18.

(a)Which numbers are common factors of 12 and 18?

(b)What is the HCF of 12 and 18?

Done

PSMila has 36 red counters and 48 blue counters. She makes identical bags. Each bag has the same number of red counters and the same number of blue counters, and no counters are left over. What is the largest number of bags she can make?

Done

PSBen has two ribbons, 24 cm and 40 cm long. He cuts both ribbons into pieces of the same length. There is no ribbon left over. What is the longest length each piece can be? Give your answer in cm.

Done

PSI am a factor of 36 and a factor of 48. I am more than 5 and less than 10. What number am I?

Done

RSam says the HCF of 12 and 30 is 12, because 12 is the bigger number. Is Sam right? Explain your answer.

Done

RAlways, sometimes or never? The HCF of two different even numbers is an even number.

Done

RWrite all the numbers less than 30 that have an HCF of 4 with 24.

Done