What is a common factor in maths?

Key points

A is a whole number which is a of two or more numbers.
The (HCF) is the greatest factor that will divide into two or more numbers.
The (LCM) is the smallest that is common to two or more numbers.
The HCF and LCM:
- are used when adding and subtracting fractions.
- can be found by listing factors and multiples. For larger values, expressing numbers as a is a more efficient method. A can also be used.

Knowing how to find factors, draw and write numbers as a product of their prime factors is useful to know when finding common factors, the HCF and LCM.

How to find common factors

A is a whole number which is a of two or more numbers. Eg, 2, 5 and 10 are common factors of 30 and 20
1 is a common factor of all whole numbers.

To find common factors:

List the factors of each number.
Compare the factors to find numbers that are the same (or common) in each list.

Examples

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Find the common factors of 30 and 20
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List the factors of 30 and the factors of 20
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Compare the lists of factors to find numbers that occur in both lists. These are the common factors. The common factors of 30 and 20 are 1, 2, 5, and 10
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Find the common factors of 24 and 25
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List the factors of 24 and the factors of 25
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Compare the lists of factors to find numbers that occur in both lists. The only common factor of 24 and 25 is 1
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Find the common factors of 12, 36 and 60
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List the factors of 12, the factors of 36 and the factors of 60
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Compare the lists of factors to find numbers that occur in all three lists. The common factors of 12, 36 and 60 are 1, 2, 3, 4, 6, and 12.12 is a factor of both 36 and 60, so all its factors will be common factors of 36 and 60

Question

What are the common factors of 40, 72 and 96?

Find the HCF of two or more numbers

The (HCF) is the greatest factor that will divide into two or more numbers.

The HCF of two or more numbers can be found by listing the factors. For larger numbers, can be used. Prime factorisation means expressing the numbers as a .

By listing:

List the factors of each number.
Compare the lists to find the greatest number common to both.

By prime factorisation:

Draw for each number. You may need to review how to draw a factor tree.
Write each number as a product of its prime factors. It can be easier to not use form.
Identify the common in each product of prime factors.
To find the HCF, multiply the common prime factors.

Remember: index form is an abbreviation for repeated multiplication. Eg, 5² means 5 × 5. 5³ means 5 × 5 × 5

Examples

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Find the highest common factor (HCF) of 30 and 20 by listing the numbers.
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List the factors of 30 and the factors of 20
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Compare the lists of factors to find the greatest number that occurs in both lists. The HCF of 30 and 20 is 10
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Find the HCF of 30 and 20 using prime factorisation.
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Draw a factor tree for 30 and a factor tree for 20. The factor trees are both complete - the prime factors are circled. Use the factor trees to write 30 and 20 as a product of their prime factors.
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Identify the common prime factors. The common prime factors of 30 and 20 are 2 and 5
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To find the HCF of 30 and 20, multiply the common prime factors. 2 x 5 = 10. The HCF of 30 and 20 is 10
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When the prime factorisation of numbers has already been completed, the HCF of the numbers can still be found. Given that 936 is 2³ × 3² × 13 and 810 is 2 × 3⁴ × 5, find the HCF of 936 and 810
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Write out the products of prime factors. It will be easier to spot the common prime factors if the products are not in index form. The common prime factors are 2, 3 and 3
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To find the HCF of 936 and 810, multiply the common prime factors. 2 x 3 x 3 = 18. The HCF of 936 and 810 is 18

Question

Find the HCF of 30 and 36

Find the LCM of two or more numbers.

A is an that is in the multiplication table of a specific number.
The (LCM) is the lowest multiple that is common to two or more numbers.

The lowest common multiple (LCM) of two of more numbers can be found by listing the numbers. For larger numbers, can be used.

By listing

List consecutive multiples for each number.
The first multiple to occur in every list is the LCM.

By prime factorisation

Draw for each number.
Write each number as a . It can be easier to not use form.
Identify the common prime factors in each product of .
The prime factors that are not common to each product are unique for that number.
To find the LCM, multiply the common factors by the unique factors of one number and the unique factors of the other number.

Examples

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Find the lowest common multiple (LCM) of 30 and 20 by listing each number’s multiples.
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List consecutive multiples of 30 and 20 until there is a matching multiple. This is the LCM. The LCM of 3 and 20 is 60
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Find the LCM of 30 and 20 using prime factorisation. This means expressing numbers as products of their prime factors.
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Draw a factor tree for 30 and a factor tree for 20. The factor trees are both complete - the prime factors are circled. Use the factor trees to write 30 and 20 as a product of their prime factors.
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The prime factors are circled. Use the factor trees to write 30 and 20 as a product of their prime factors. 30 = 2 × 3 × 5 and 20 = 2 × 2 × 5. The common factors are the factors that occur in the prime factorisation of both 30 and 20 - these are shown in orange. The unique factors are those that are unique to 30 (shown in blue) and unique to 20 (shown in purple).
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30 and 20 have the prime factors 2 and 5 in common. The unique prime factor of 30 is 3. The unique prime of 20 is 2. These are needed to find the LCM.
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To find the LCM of 30 and 20, multiply the common prime factors by the unique prime factor of 30 (3) and the unique prime factor of 20 (2). 2 × 5 × 3 × 2 = 60. The LCM of 30 and 20 is 60
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When the prime factorisation of numbers has already been completed, the LCM of the numbers can still be found. Given that 936 is 2³ × 3² × 13 and 810 is 2 × 3⁴ × 5, find the LCM of 936 and 810
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Write the products of prime factors without index notation. This makes it easier to spot the common prime factors. The common prime factors are 2, 3 and 3
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To find the LCM of 936 and 810, multiply the common prime factors by the unique prime factors of 936 (2, 2 and 13) and the unique prime factors of 810 (3, 3, 5). 2 × 3 × 3 × 2 × 2 × 13 × 3 × 3 × 5 = 42,120. The LCM of 936 and 810 is 42,120

Question

Find the LCM of 30 and 36

How to use a Venn diagram to find the HCF and LCM

To use a to find the HCF and LCM of two numbers:

Write each number as a without using index form.
Draw a Venn diagram:

Draw a circle for each number.
Place the factors that are common to both numbers in the intersection (the overlap) of the circles.
Place the remaining prime factors for the first number in its circle.
Place the remaining for the second number in its circle.
Check that each circle contains the full for each number.

To find the HCF of the numbers - multiply the numbers in the intersection.
To find the LCM of the numbers - multiply all the numbers in the circles in the Venn diagram.

Examples

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Find the HCF and LCM of 30 and 20 using a Venn diagram.
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Write each number as a product of its prime factors. 30 = 2 x 3 x 5. 20 = 2 x 2 x 5. Then, draw the Venn diagram. Draw one circle for 30 and another for 20
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The common prime factors of 30 and 20 is 2 and 5. Place these in the intersection of the Venn diagram. To find the HCF of 30 and 20, multiply the common prime factors. 2 x 5 = 10. The HCF of 30 and 20 is 10
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Place the remaining prime factor for 30 (3) in its circle and the remaining prime factor for 20 (2) in its circle.
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Check that each circle contains all the prime factors in the product of prime factors for each number.
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Look at the whole Venn diagram. Multiply all the prime factors in the diagram. The LCM of 30 and 20 is 60

Question

Use the Venn diagram to find the HCF and LCM of 240 and 180

A diagram of a box with two circles overlapping. The right circle is labelled one-hundred and eighty and the centre is labelled three; the left circle is labelled two-hundred and forty and the centre is labelled two and two. The overlap is labelled two, two, three and five.

Practise common factors

Practise common factors and multiples in this quiz. You may need a pen and paper to complete some of these questions.

Quiz

Real-world maths

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Planning events to maximise efficiency of costs.

The HCF and LCM can be useful for a variety of businesses when ordering materials and stock.

Calculating the HCF can help an independent stationery business to use their materials efficiently – for example, when making cards. Card stock can be bought in lengths of 24 inches and 36 inches. The HCF of 24 and 36 is 12. By making cards at a size of 12 inches, they can limit the amount of waste produced from card stock.

Calculating the LCM can allow an event planner to manage their budget effectively. For example, they may want to order the same amount of stickers and tote bags to include in gift bags. If the stickers are sold in packs of 15 and gift bags are sold in packs of 6, they should buy these items in batches of 30. The LCM of 15 and 6 is 30. This means that 2 packs of stickers and 5 packs of tote bags gives 30 of each gift. This way, the event planner has matching amounts to put in the gift bags.

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