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Maths · Number

HCF and LCM

Every whole number is built from prime numbers in exactly one way. Once you have a number's prime factors, the highest common factor and lowest common multiple of two numbers take seconds to find.

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Last Lesson and Before

Answer each one, then check.

  1. 1

    List all the factors of 12.

    Show answerHide answer

    1, 2, 3, 4, 6, 12

  2. 2

    Write down the first five multiples of 6.

    Show answerHide answer

    6, 12, 18, 24, 30

  3. 3

    Is 51 a prime number?

    Show answerHide answer

    No: \(51 = 3 \times 17\).

  4. 4

    What is \(2^3\)?

    Show answerHide answer

    \(2 \times 2 \times 2 = 8\)

  5. 5

    Last lesson: round 0.004 76 to 2 significant figures.

    Show answerHide answer

    \(0.0048\)

Learning Objectives

  1. 1Know the difference between factors, multiples and prime numbers.
  2. 2Write a number as a product of its prime factors, in index form.
  3. 3Find the HCF and LCM of two numbers using prime factors.
  4. 4Use a Venn diagram of prime factors to find the HCF and LCM.
  5. 5Decide whether a problem needs the HCF or the LCM, and solve it.

Factors, Multiples and Primes

  • 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, ...

  • 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.

  • 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: \(2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5\).

  • Check

    Multiply your answer out: it should give the number you started with.

Writing a Number as a Product of Primes

Write 360 as a product of its prime factors. Give your answer in index form.

Show the solutionHide the solution
  1. 1 Split into two factors \(360 = 10 \times 36\)
  2. 2 Keep splitting until every branch is prime \(10 = 2 \times 5\), \(36 = 4 \times 9 = 2 \times 2 \times 3 \times 3\)
  3. 3 Collect the primes, smallest first \(2 \times 2 \times 2 \times 3 \times 3 \times 5\)
  4. 4 Write repeats as powers \(2^3 \times 3^2 \times 5\)
  5. 5 Check \(8 \times 9 \times 5 = 360\)

Answer\(360 = 2^3 \times 3^2 \times 5\)

HCF or LCM?

Highest Common Factor (HCF)

  • 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.

Lowest Common Multiple (LCM)

  • 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

Find the HCF and the LCM of 84 and 120.

Show the solutionHide the solution
  1. 1 Write each number as a product of primes \(84 = 2^2 \times 3 \times 7\) and \(120 = 2^3 \times 3 \times 5\)
  2. 2 HCF: take each shared prime at its lower power \(2^2 \times 3\)
  3. 3 Work it out HCF \(= 12\)
  4. 4 LCM: take every prime at its higher power \(2^3 \times 3 \times 5 \times 7\)
  5. 5 Work it out LCM \(= 840\)

AnswerHCF \(= 12\), LCM \(= 840\)

A Venn Diagram of Prime Factors

84 = 2 × 2 × 3 × 7

  • 7

120 = 2 × 2 × 2 × 3 × 5

  • 2
  • 5

Shared primes

  • 2
  • 2
  • 3

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: \(2 \times 2 \times 3 = 12\).

  • LCM

    Multiply every number in the diagram: \(7 \times 2 \times 2 \times 3 \times 2 \times 5 = 840\).

  • Check

    HCF \(\times\) LCM = the two numbers multiplied: \(12 \times 840 = 10\,080 = 84 \times 120\).

  • 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.

Spot the Clue

  • 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.

  • 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

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?

Show the solutionHide the solution
  1. 1 "Leave at the same time again" means the LCM LCM of 12 and 18
  2. 2 Prime factors \(12 = 2^2 \times 3\) and \(18 = 2 \times 3^2\)
  3. 3 Highest power of each prime \(2^2 \times 3^2 = 36\)
  4. 4 Add 36 minutes to 9:00 am 9:36 am

Answer9:36 am

A Highest Common Factor Problem

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?

Show the solutionHide the solution
  1. 1 "Largest number of identical packs" means the HCF HCF of 48 and 60
  2. 2 Prime factors \(48 = 2^4 \times 3\) and \(60 = 2^2 \times 3 \times 5\)
  3. 3 Lower power of each shared prime \(2^2 \times 3 = 12\)
  4. 4 Share each item between 12 packs \(48 \div 12 = 4\), \(60 \div 12 = 5\)

Answer12 packs, each with 4 pencils and 5 rubbers

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 \(13 \times 17 = 221\), 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.

13 and 17 Years between emergences - both prime
221 Years between joint emergences: LCM(13, 17) = 13 × 17

HCF or LCM?

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...?

  1. 1Explain what factors, multiples and primes are.
  2. 2Draw a factor tree.
  3. 3Write a number as a product of primes in index form.
  4. 4Find the HCF using prime factors.
  5. 5Find the LCM using prime factors.
  6. 6Use a Venn diagram of prime factors.
  7. 7Recognise HCF and LCM word problems.
  8. 8Answer the question that was actually asked.

Summary & Exam Focus

  • 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.

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) (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.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Factor
A number that divides exactly into another number.
Multiple
A number in another number's times table.
Prime number
A number with exactly two factors, 1 and itself.
Prime factor decomposition
Writing a number as a product of its prime factors.
Index form
Writing repeated factors as powers, e.g. \(2^3 \times 3^2\).
Highest common factor (HCF)
The largest number that is a factor of two or more numbers.
Lowest common multiple (LCM)
The smallest number that is a multiple of two or more numbers.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    Write 180 as a product of its prime factors. Give your answer in index form.

    Show answerHide answer

    Model answer

    \(180 = 2^2 \times 3^2 \times 5\)

    Mark scheme

    • A correct method, e.g. a factor tree with at least two correct steps, or 2, 2, 3, 3, 5 seen — M1
    • \(2^2 \times 3^2 \times 5\) — A1
  2. Question 2 Non-calculator 2 marks

    \(A = 2^3 \times 3 \times 5^2\) and \(B = 2^2 \times 3^3 \times 7\). (a) Find the highest common factor of \(A\) and \(B\). (b) Find the lowest common multiple of \(A\) and \(B\). You may leave your answers in index form.

    Show answerHide answer

    Model answer

    (a) HCF \(= 2^2 \times 3 = 12\) (b) LCM \(= 2^3 \times 3^3 \times 5^2 \times 7 = 37\,800\)

    Mark scheme

    • (a) \(2^2 \times 3\) or \(12\) — B1
    • (b) \(2^3 \times 3^3 \times 5^2 \times 7\) or \(37\,800\) — B1
  3. Question 3 Non-calculator 3 marks

    Light A flashes every 20 seconds. Light B flashes every 45 seconds. Both lights flash at 10 pm. At what time will they next flash together?

    Show answerHide answer

    Model answer

    \(20 = 2^2 \times 5\) and \(45 = 3^2 \times 5\), so the LCM is \(2^2 \times 3^2 \times 5 = 180\) seconds = 3 minutes. They next flash together at 10:03 pm.

    Mark scheme

    • Lists multiples of 20 and 45, or writes both as products of primes — P1
    • 180 (seconds) or 3 minutes — P1
    • 10:03 pm — A1
  4. Question 4 Non-calculator 3 marks

    Paper cups are sold in packs of 24. Paper plates are sold in packs of 36. Amir wants to buy exactly the same number of cups and plates. What is the smallest number of packs of cups and the smallest number of packs of plates he can buy?

    Show answerHide answer

    Model answer

    \(24 = 2^3 \times 3\) and \(36 = 2^2 \times 3^2\), so the LCM is \(2^3 \times 3^2 = 72\). \(72 \div 24 = 3\) packs of cups and \(72 \div 36 = 2\) packs of plates.

    Mark scheme

    • Lists multiples of 24 and 36, or writes both as products of primes — P1
    • \(72\) — P1
    • 3 packs of cups and 2 packs of plates — A1
  5. Question 5 Non-calculator 2 marks

    The HCF of two numbers is 6. The LCM of the same two numbers is 36. Both numbers are greater than 6. Find the two numbers.

    Show answerHide answer

    Model answer

    Both numbers are multiples of 6 and factors of 36, so they come from 12 and 18 (6 and 36 are ruled out). HCF(12, 18) = 6 and LCM(12, 18) = 36. The numbers are 12 and 18.

    Mark scheme

    • Considers multiples of 6 that are factors of 36, e.g. 12, 18 seen — P1
    • 12 and 18 — A1

Quick check

  1. Which of these is 60 written as a product of its prime factors?

    1. A\(6 \times 10\)
    2. B\(2^2 \times 3 \times 5\)
    3. C\(2 \times 30\)
    4. D\(4 \times 15\)
    Show answerHide answer

    B: \(2^2 \times 3 \times 5\)

    \(2 \times 2 \times 3 \times 5 = 60\), and every factor is prime.

  2. What is the highest common factor of 18 and 24?

    1. A2
    2. B3
    3. C6
    4. D72
    Show answerHide answer

    C: 6

    \(18 = 2 \times 3^2\) and \(24 = 2^3 \times 3\). Shared: \(2 \times 3 = 6\).

  3. Two bells ring every 15 minutes and every 25 minutes. They ring together at noon. When do they next ring together?

    1. A12:40 pm
    2. B12:25 pm
    3. C1:00 pm
    4. D1:15 pm
    Show answerHide answer

    D: 1:15 pm

    LCM of 15 and 25: \(3 \times 5^2 = 75\) minutes after noon, which is 1:15 pm.

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