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1.4 Factors and Multiples

GCSE Maths Foundation · Edexcel · Chapter 1: Number

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: factors, multiples, prime numbers, HCF and LCM by listing, and real-life problems. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Find all the factors of a number using factor pairs, and list multiples
  • Recognise prime numbers and use simple tests for divisibility
  • Find the highest common factor (HCF) and lowest common multiple (LCM) of two numbers
  • Decide whether a problem needs the HCF or the LCM and solve it

Key words

  • Factor: A number that divides into another exactly. 4 is a factor of 12.
  • Multiple: A number in the times table of another. 12 is a multiple of 4.
  • Prime number: A number with exactly two factors: 1 and itself. 2, 3, 5, 7, 11, … (1 is not prime).
  • HCF: Highest common factor: the biggest number that is a factor of both numbers.
  • LCM: Lowest common multiple: the smallest number that is a multiple of both numbers.

Factors and factor pairs

Factors come in pairs that multiply to give the number. Start at 1 and work upwards until the pairs start to repeat.

Worked example: all the factors of 36

1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6 (5 is not a factor)
The next pair would repeat, so stop.

Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36

Multiples and divisibility

The multiples of 7 are 7, 14, 21, 28, 35, … They go on for ever. These quick tests tell you whether a number divides exactly.

  • 2; Test last digit is even; Example 3 578
  • 3; Test digit sum is a multiple of 3; Example 471: 4 + 7 + 1 = 12
  • 4; Test last two digits make a multiple of 4; Example 2 316: 16 = 4 × 4
  • 5; Test last digit is 0 or 5; Example 8 435
  • 6; Test passes the tests for 2 and for 3; Example 1 362
  • 9; Test digit sum is a multiple of 9; Example 5 724: 5 + 7 + 2 + 4 = 18
  • 10; Test last digit is 0; Example 960

Worked example: is it a multiple?

Is 91 a multiple of 7?
7 × 13 = 91, so yes.

Answer: yes, 91 = 7 × 13

Prime numbers

Learn: The primes less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. 2 is the only even prime. 1 is not prime because it has only one factor.

Worked example: spotting a prime

Which of 51, 53 and 57 is prime?
51 = 3 × 17 and 57 = 3 × 19 (digit sums 6 and 12 are multiples of 3).
53 is not divisible by 2, 3, 5 or 7, so it is prime.

Answer: 53

HCF and LCM by listing

For the HCF, list the factors of both numbers and pick the biggest one in both lists. For the LCM, list multiples of the bigger number until you reach one the smaller number divides into.

Worked example: LCM

Find the LCM of 6 and 8.
Multiples of 8: 8, 16, 24 … 24 is the first that 6 divides.

Answer: LCM = 24

HCF or LCM? Solving problems

LCM questions are about things happening again at the same time (buses, lights, packs to buy). HCF questions are about splitting things into the largest equal groups.

Worked example: buses (LCM)

Bus A leaves every 12 minutes, bus B every 15 minutes. Both leave at 8:00 am. When do they next leave together?
Multiples of 15: 15, 30, 45, 60. 60 is the first multiple of 12 too.

Answer: after 60 minutes, at 9:00 am

Worked example: fruit bags (HCF)

24 apples and 36 oranges are shared into identical bags with nothing left over. What is the greatest number of bags?
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The biggest that also divides 36 is 12.
Each bag has 24 ÷ 12 = 2 apples and 36 ÷ 12 = 3 oranges.

Answer: 12 bags

Common mistakes

  • Mixing up factors and multiples: factors of 12 are 12 or less, multiples are 12 or more.
  • Saying 1 is a prime number. A prime has exactly two factors.
  • Thinking every odd number is prime: 9, 15, 21 and 27 are odd but not prime.
  • Multiplying the two numbers to get the LCM: 6 × 8 = 48, but the LCM of 6 and 8 is 24.

Revision questions

Try these, then check your answers below each one.

Question 1. Which of these numbers is a factor of 56? (Choose from: 6, 8, 12, 16.)

Answer: 8

Working: 56 ÷ 8 = 7 exactly. 6, 12 and 16 all leave a remainder.

Question 2. Find the highest common factor (HCF) of 24 and 60.

Answer: 12

Working: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. 12 is also a factor of 60 (60 ÷ 12 = 5), and no bigger factor of 24 divides 60. HCF = 12.

Question 3. Find the lowest common multiple (LCM) of 4 and 10.

Answer: 20

Working: Multiples of 10: 10, 20, … 20 is the first that is also a multiple of 4. LCM = 20.

Related lessons

Continue in the full course

This lesson is section 1.4 (EGCMF0104) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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