EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
HCF and LCM
Number · Lesson 3 of 7
Last Lesson and Before
Answer each one, then check.
1. List all the factors of 12.
1, 2, 3, 4, 6, 12
2. Write down the first five multiples of 6.
6, 12, 18, 24, 30
3. Is 51 a prime number?
No: 51 = 3 × 17.
4. What is 2³?
2 × 2 × 2 = 8
5. Last lesson: round 0.004 76 to 2 significant figures.
0.0048
Learning Objectives
1. Know the difference between factors, multiples and prime numbers.
2. Write a number as a product of its prime factors, in index form.
3. Find the HCF and LCM of two numbers using prime factors.
4. Use a Venn diagram of prime factors to find the HCF and LCM.
5. Decide whether a problem needs the HCF or the LCM, and solve it.
Factors, Multiples and Primes
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Factor A number that divides exactly into another. The factors of 10 are 1, 2, 5 and 10. |
Multiple A number in another number's times table. The multiples of 10 are 10, 20, 30, ... |
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Prime number A number with exactly two factors: 1 and itself. 2, 3, 5, 7, 11, 13, ... |
1 is not prime It has only one factor. |
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2 is the only even prime Every other even number has 2 as a factor. |
Prime factor A factor that is also a prime number. The prime factors of 10 are 2 and 5. |
Products of Prime Factors
Every whole number greater than 1 can be written as a product of prime numbers in exactly one way.
▸ Factor tree. Split the number into any two factors, then keep splitting until every branch ends in a prime.
▸ Circle the primes. Circle each prime as you reach it, so none is missed.
▸ Index form. Write repeated primes as powers, smallest prime first: 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
▸ Check. Multiply your answer out: it should give the number you started with.
A Factor Tree for 360
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It does not matter how you start: 360 = 10 × 36, 360 = 2 × 180 and 360 = 12 × 30 all end in the same set of primes. That is why the prime factors of a number are like its fingerprint. |
360 = 2³ × 3² × 5 |
Writing a Number as a Product of Primes
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Write 360 as a product of its prime factors. Give your answer in index form. |
1. Split into two factors
360 = 10 × 36
2. Keep splitting until every branch is prime
10 = 2 × 5, 36 = 4 × 9 = 2 × 2 × 3 × 3
3. Collect the primes, smallest first
2 × 2 × 2 × 3 × 3 × 5
4. Write repeats as powers
2³ × 3² × 5
5. Check
8 × 9 × 5 = 360
Answer: 360 = 2³ × 3² × 5
PART ONE
HCF and LCM
Using prime factors.
HCF or LCM?
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HIGHEST COMMON FACTOR (HCF) |
LOWEST COMMON MULTIPLE (LCM) |
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▸ The biggest number that divides into both numbers. ▸ It is never bigger than the smaller number. ▸ Prime factors: multiply the primes the numbers SHARE. ▸ Use the LOWER power of each shared prime. ▸ Used for splitting things into equal groups. |
▸ The smallest number both numbers divide into. ▸ It is never smaller than the larger number. ▸ Prime factors: multiply EVERY prime that appears. ▸ Use the HIGHER power of each prime. ▸ Used for when things happen together again. |
HCF and LCM from Prime Factors
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Find the HCF and the LCM of 84 and 120. |
1. Write each number as a product of primes
84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5
2. HCF: take each shared prime at its lower power
2² × 3
3. Work it out
HCF = 12
4. LCM: take every prime at its higher power
2³ × 3 × 5 × 7
5. Work it out
LCM = 840
Answer: HCF = 12, LCM = 840
A Venn Diagram of Prime Factors
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84 = 2 × 2 × 3 × 7 |
SHARED PRIMES |
120 = 2 × 2 × 2 × 3 × 5 |
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▸ 7 |
▸ 2 ▸ 2 ▸ 3 |
▸ 2 ▸ 5 |
Reading the Venn Diagram
Put the shared primes in the overlap and the rest in their own circles.
▸ HCF. Multiply the numbers in the overlap: 2 × 2 × 3 = 12.
▸ LCM. Multiply every number in the diagram: 7 × 2 × 2 × 3 × 2 × 5 = 840.
▸ Check. HCF × LCM = the two numbers multiplied: 12 × 840 = 10 080 = 84 × 120.
▸ Careful. Each prime goes in the diagram once for each time it is shared - two 2s are shared here, so two 2s go in the middle.
PART TWO
Word Problems
Is it the HCF or the LCM?
Spot the Clue
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LCM: "at the same time again" Buses, lights or events that repeat, meeting again. |
LCM: "the same number of each" Buying packs of different sizes to get equal numbers. |
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HCF: "the largest possible" The biggest group size, tile or length that fits exactly. |
HCF: "share equally with none left over" Splitting two amounts into identical groups. |
A Lowest Common Multiple Problem
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Bus A leaves the station every 12 minutes. Bus B leaves every 18 minutes. Both buses leave at 9:00 am. When do they next leave at the same time? |
1. "Leave at the same time again" means the LCM
LCM of 12 and 18
2. Prime factors
12 = 2² × 3 and 18 = 2 × 3²
3. Highest power of each prime
2² × 3² = 36
4. Add 36 minutes to 9:00 am
9:36 am
Answer: 9:36 am
A Highest Common Factor Problem
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A teacher has 48 pencils and 60 rubbers. She makes identical packs using all of them. What is the largest number of packs she can make, and what is in each pack? |
1. "Largest number of identical packs" means the HCF
HCF of 48 and 60
2. Prime factors
48 = 2⁴ × 3 and 60 = 2² × 3 × 5
3. Lower power of each shared prime
2² × 3 = 12
4. Share each item between 12 packs
48 ÷ 12 = 4, 60 ÷ 12 = 5
Answer: 12 packs, each with 4 pencils and 5 rubbers
Case Study
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CASE STUDY Prime-Numbered Cicadas Periodical cicadas in North America spend most of their lives underground and come out in their millions only once every 13 or 17 years - both prime numbers. One idea is that a prime cycle makes it hard for predators with shorter cycles to keep meeting them: a predator on a 2, 3, 4 or 6 year cycle rarely lines up with a 13 or 17 year one. Because 13 and 17 share no factors, their LCM is 13 × 17 = 221, so a 13-year brood and a 17-year brood living side by side come out together only once every 221 years. In 2024 two neighbouring broods did exactly that in Illinois - the first time since 1803. |
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13 and 17 Years between emergences - both prime |
221 Years between joint emergences: LCM(13, 17) = 13 × 17 |
Key Terms
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Factor A number that divides exactly into another number. |
Multiple A number in another number's times table. |
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Prime number A number with exactly two factors, 1 and itself. |
Prime factor decomposition Writing a number as a product of its prime factors. |
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Index form Writing repeated factors as powers, e.g. 2³ × 3². |
Highest common factor (HCF) The largest number that is a factor of two or more numbers. |
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Lowest common multiple (LCM) The smallest number that is a multiple of two or more numbers. |
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Your Task: HCF or LCM?
12 minutes
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For each problem, decide whether it needs the HCF or the LCM, then solve it. (a) Two lighthouses flash every 20 seconds and every 45 seconds. They flash together now. How long until they next flash together? (b) A rectangular floor 240 cm by 180 cm is covered with identical square tiles with no cutting. What is the largest tile that can be used? (c) Hot dogs come in packs of 8 and rolls in packs of 10. What is the least number of each pack needed to have the same number of hot dogs and rolls? 1. Decide: HCF or LCM? 2. Write both numbers as products of primes, or list multiples. 3. Answer the question that was asked. |
A good answer shows: (a) LCM = 180 seconds (3 minutes). (b) HCF = 60 cm tiles. (c) LCM = 40: 5 packs of hot dogs and 4 packs of rolls.
Can I...?
☐ Explain what factors, multiples and primes are.
☐ Draw a factor tree.
☐ Write a number as a product of primes in index form.
☐ Find the HCF using prime factors.
☐ Find the LCM using prime factors.
☐ Use a Venn diagram of prime factors.
☐ Recognise HCF and LCM word problems.
☐ Answer the question that was actually asked.
Summary
✓ Every number has one set of prime factors: use a factor tree and write it in index form.
✓ HCF: shared primes at the lower power - the overlap of the Venn diagram.
✓ LCM: every prime at the higher power - everything in the Venn diagram.
✓ HCF problems split things into equal groups; LCM problems ask when things happen together again.
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EXAM FOCUS Paper cups are sold in packs of 24. Paper plates are sold in packs of 36. Amir wants exactly the same number of cups and plates. What is the smallest number of packs of each he can buy? (3 marks) After finding the HCF or LCM, reread the question. It often asks for something you work out from it - a time, a number of packs, a group size. |