GCSE Maths · Factors, Multiples and Primes · Foundation · Lesson 2.4

GCSE Maths: Highest common factor and lowest common multiple (Foundation)

Part of the GCSE Maths factors, multiples and primes chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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2.4

Highest common factor and lowest common multiple

Edexcel 1MA1: N4
Grade 3-7

In Section 2.1 you met common factors and common multiples. Two special ones are used a lot.

The highest common factor (HCF) of two numbers is the biggest number that divides into both. We use it to cut things into the largest equal pieces, or to simplify fractions (Chapter 3).

The lowest common multiple (LCM) is the smallest number that is in both times tables. We use it when two things repeat and we want to know when they line up, or to find a common denominator.

Method 1: listing. For the HCF, list the factors of each number and find the biggest number in both lists. For the LCM, list the multiples of each number until you find the first one in both lists.

Method 2: prime factors and a Venn diagram. This method works for big numbers where lists would be long. Write each number as a product of primes. Draw two overlapping circles, one for each number. Put the primes that both numbers share in the middle, and the leftover primes in the outer parts.

12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Both share a 2 and a 3 (the middle). 12 has one more 2 and 18 has one more 3. HCF = 2 × 3 = 6. LCM = 2 × 3 × 2 × 3 = 36.

HCF = multiply the numbers in the middle. LCM = multiply all the numbers in the diagram (the middle ones only once). Why? The middle holds what both numbers have in common, which builds the biggest shared factor. The LCM has to contain everything that either number needs, and using the shared primes only once keeps it as small as possible.

60 = 2 × 2 × 3 × 5 and 84 = 2 × 2 × 3 × 7. The middle is 2, 2, 3 so HCF = 12. All the numbers multiplied: 2 × 2 × 3 × 5 × 7 = 420 = LCM.

Which one do I need? If the answer must be smaller than or equal to both numbers (cutting, sharing, grouping), it is the HCF. If it is bigger than or equal to both (meeting again, repeating, buying enough), it is the LCM.

Key ideas

HCF = the biggest number that divides into both. LCM = the smallest number in both times tables.
Listing works for small numbers. For larger numbers use prime factors and a Venn diagram.
Venn diagram: HCF = multiply the middle; LCM = multiply everything in the diagram.
HCF ≤ both numbers; LCM ≥ both numbers.
Check: HCF × LCM = the two numbers multiplied together.

Worked example 2.4.1

Find the HCF of 24 and 36 by listing factors.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. The highest common factor (HCF) is the biggest number that divides into both 24 and 36. Here we list the factors.

List the factors of 24. These are all the numbers that divide into it exactly.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
List the factors of 36. Do the same for the other number.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Find the common factors. These are the numbers that appear in both lists.
Common factors: 1, 2, 3, 4, 6, 12
Pick the highest one. The highest common factor is the biggest number in the common list.
HCF = 12

Check. 24 ÷ 12 = 2 and 36 ÷ 12 = 3, and 2 and 3 have no common factor except 1, so no bigger number works. ✓

Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.

Worked example 2.4.2

Find the LCM of 6 and 8 by listing multiples.

Method. The lowest common multiple (LCM) is the smallest number that is in both the 6 times table and the 8 times table. Here we list the multiples.

List the multiples of 6. Count up in 6s. Keep going until you have passed a number that is also in the other list.
Multiples of 6: 6, 12, 18, 24, ...
List the multiples of 8. Count up in 8s in the same way.
Multiples of 8: 8, 16, 24, ...
Find the first number that is in both lists. That is the lowest common multiple.
LCM = 24

Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3, both whole numbers. And no smaller number appears in both lists. ✓

Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.

Worked example 2.4.3

Find the HCF and the LCM of 60 and 84.

Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 60 and 84.

Write 60 and 84 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
60 = 2 × 2 × 3 × 5
84 = 2 × 2 × 3 × 7
Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.
HCF = 2 × 2 × 3 = 12
LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.
LCM = 5 × 2 × 2 × 3 × 7 = 420

Check. HCF × LCM = 12 × 420 = 5040 and 60 × 84 = 5040. They are equal. ✓

Watch out. Check you give both answers, labelled HCF and LCM.

Worked example 2.4.4

Bus A leaves the station every 12 minutes. Bus B leaves every 18 minutes. Both leave at 9:00 am. At what time do they next leave together?

Method. The lowest common multiple (LCM) is the smallest number that is in both the 12 times table and the 18 times table. Here we use prime factors in a Venn diagram.

Decide what the question is asking. The buses leave together again after a number of minutes that is in both the 12 times table and the 18 times table. We want the first time, so we need the lowest common multiple.
Find the lowest common multiple of 12 and 18.
Write 12 and 18 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
12 = 2 × 2 × 3
18 = 2 × 3 × 3
Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.
LCM = 2 × 2 × 3 × 3 = 36
Answer the question in words. Always finish by saying what the number means.
They leave together again after 36 minutes, which is 9:36 am.

Check. HCF × LCM = 6 × 36 = 216 and 12 × 18 = 216. They are equal. ✓

Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.

Watch out

Do not mix the two up: HCF is a factor (small), LCM is a multiple (big). And the LCM is not always the two numbers multiplied together: LCM of 6 and 8 is 24, not 48.

Exercise 2D HCF and LCM

Non-calculator

FFind the HCF of each pair of numbers by listing their factors.

(a)12 and 18

(b)8 and 20

(d)16 and 40

(e)14 and 21

(f)24 and 36

Done

FFind the LCM of each pair of numbers by listing their multiples.

(a)3 and 4

(b)6 and 8

(d)4 and 18

(e)9 and 12

(f)5 and 7

Done

FFind the HCF and the LCM of each pair. Use listing.

(a)4 and 6

(b)6 and 9

(d)10 and 25

Done

FWrite each number as a product of primes, then use a Venn diagram to find the HCF and LCM.

(a)18 and 24

(b)30 and 42

(d)40 and 56

(e)45 and 75

(f)60 and 90

Done

FFind the HCF and LCM of each pair of numbers.

(a)84 and 126

(b)72 and 108

(d)168 and 252

Done

PSTwo buses leave a depot together at 8:00 am. One leaves every 15 minutes and the other every 20 minutes. At what time do they next leave together?

Done

PSA ribbon 84 cm long and a ribbon 126 cm long are cut into equal pieces with no ribbon left over. All the pieces must be the same length. What is the greatest possible length of each piece?

Done

PSPencils come in packs of 12. Rubbers come in packs of 15. Ravi wants to buy the same number of pencils as rubbers. What is the smallest number of each pack he must buy?

Done

PSTwo lighthouses flash together at 8:00 pm. One flashes every 20 seconds and the other every 30 seconds. How many more times in the next 5 minutes do they flash together?

Done

PSA baker has 48 chocolate biscuits and 72 oat biscuits. She makes identical bags, using all the biscuits. Each bag has the same number of chocolate biscuits and the same number of oat biscuits. What is the greatest number of bags she can make, and how many of each biscuit are in a bag?

Done

PSThe Venn diagram shows the prime factors of two numbers, A and B.

(a)Work out A.

(b)Work out B.

(c)Work out the HCF of A and B.

(d)Work out the LCM of A and B.

Done

RSara says: “The LCM of two numbers is always bigger than both of them.” Is she correct? Give an example to explain.

Done

R(a) Find the HCF of 8 and 15. (b) Find the LCM of 8 and 15. (c) What do you notice about the LCM and 8 × 15?

(a)(a)

(b)(b)

(c)(c)

Done

PSAlice visits the library every 6 days, Ben every 8 days and Cara every 9 days. All three are there on Monday 1 May. After how many days will they next all be there together?

Done

RTwo different whole numbers are both between 10 and 40. Their HCF is 5 and their LCM is 60. Find the two numbers.

Done