GCSE Prime factorisation Practice Questions

Free GCSE Prime factorisation practice questions with full step-by-step worked solutions. Covers prime factorisation, factor tree, index form, evaluating index form. Practise exam-style problems and check your method.

prime factorisationfactor treeindex formevaluating index formcheckingHCF from prime factors
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
Write 1212 as a product of its prime factors.
Show worked solution

Worked solution

  1. Split 12 into a factor pair

    12=4×312=4\times3

    Start a factor tree by breaking 12 into any two factors.

  2. Break down until all prime

    4=2×24=2\times2

    44 splits into 2×22 \times 2, and 33 is already prime.

  3. Write in index form

    12=22×312=2^2\times3

    Two 22s become 222², giving 12=22×312 = 2² \times 3.

Answer
22×32^2\times3
Question 2
1 markeasy
Write 2×2×2×32\times2\times2\times3 in index form.
Show worked solution

Worked solution

  1. Count the 2s

    2×2×2=232\times2\times2=2^3

    Three 22s multiplied together is 232³.

  2. The 3 appears once

    33

    There is a single 3.

  3. Write in index form

    23×32^3\times3

    So 2×2×2×3=23×32 \times 2 \times 2 \times 3 = 2³ \times 3.

Answer
23×32^3\times3
Question 3
2 marksintermediate
Work out the value of 23×322^3\times3^2.
Show worked solution

Worked solution

  1. Work out the first power

    23=82^3=8

    23=2×2×2=82³ = 2 \times 2 \times 2 = 8.

  2. Work out the second power

    32=93^2=9

    32=3×3=93² = 3 \times 3 = 9.

  3. Multiply the results

    8×98\times9

    Now multiply 8 by 9.

  4. Work it out

    8×9=728\times9=72

    8×9=728 \times 9 = 72.

  5. So the value is

    7272

    The value is 72.

  6. State the answer

    7272

    So 23×32=722³ \times 3² = 72.

Answer
7272
Question 4
4 markshard
Find the HCF and LCM of 9696 and 144144 using their prime factorisations.
Show worked solution

Worked solution

  1. Factorise 96

    96=25×396=2^5\times3

    Write 96 as a product of primes.

  2. Factorise 144

    144=24×32144=2^4\times3^2

    Write 144 as a product of primes.

  3. HCF rule

    common, lowest power\text{common, lowest power}

    Take each shared prime to its lowest power.

  4. Shared primes

    2 and 32\ \text{and}\ 3

    Both contain 2 and 3.

  5. Lowest powers

    24×32^4\times3

    Lowest power of 2 is 2⁴, of 3 is 3¹.

  6. Work out the HCF

    16×3=4816\times3=48

    So the HCF is 48.

  7. LCM rule

    each prime, highest power\text{each prime, highest power}

    Take each prime to its highest power.

  8. Highest powers

    25×322^5\times3^2

    252⁵ (from 9696) and 323² (from 144144).

  9. Work out the LCM

    32×9=28832\times9=288

    So the LCM is 288.

  10. State the answers

    HCF=48, LCM=288\text{HCF}=48,\ \text{LCM}=288

    So HCF = 48 and LCM = 288.

Answer
HCF = 48, LCM = 288
Question 5
6 markschallenging
Explain why prime factorisation gives a reliable method for finding the HCF and LCM even of large numbers, where listing all factors would be slow.
Show worked solution

Worked solution

  1. Every number factorises uniquely

    one prime factorisation\text{one prime factorisation}

    Each whole number has exactly one prime factorisation.

  2. Listing all factors is slow

    many factors\text{many factors}

    A large number can have very many factors.

  3. And error-prone

    easy to miss one\text{easy to miss one}

    It is easy to miss a factor or list one twice.

  4. Prime factorisation is compact

    a few prime powers\text{a few prime powers}

    The factorisation records everything with just a few prime powers.

  5. For the HCF, use shared primes

    lowest powers\text{lowest powers}

    Take each shared prime to its lowest power.

  6. This gives the HCF directly

    HCF\text{HCF}

    No listing of factors is needed.

  7. For the LCM, use highest powers

    highest powers\text{highest powers}

    Take each prime to its highest power.

  8. This gives the LCM directly

    LCM\text{LCM}

    Again, no long list is needed.

  9. You only handle a few numbers

    compare exponents\text{compare exponents}

    You just compare exponents, not dozens of factors.

  10. So it scales to large numbers

    works for big numbers\text{works for big numbers}

    The method stays quick even when the numbers are large.

  11. Example numbers

    84=22×3×784=2^2\times3\times7

    Take 84, for instance.

  12. And another

    360=23×32×5360=2^3\times3^2\times5

    And 360.

  13. HCF quickly

    22×3=122^2\times3=12

    Lowest shared powers give HCF = 12 straight away.

  14. LCM quickly

    23×32×5×7=25202^3\times3^2\times5\times7=2520

    Highest powers give LCM = 2520 without listing factors.

  15. State the point

    unique factorisation\text{unique factorisation}

    Because prime factorisation is unique and compact, it gives a reliable, fast way to find HCF and LCM even for large numbers.

Answer
Prime factorisation is unique, so you compare prime powers (lowest for HCF, highest for LCM) instead of listing every factor — fast and reliable even for large numbers

Unlock 65 more Prime factorisation questions

Create a free account to work through every GCSE Prime factorisation question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Prime factorisation practice

Hard QuestionsWorked Solutions

Related Number topics

Upper and lower boundsError intervalsEstimating powers and rootsEstimation and checkingExact calculationFactors, multiples and primes