Factor (Divisor): A whole number that divides another number exactly (no remainder).

E.g. 3 is a factor of 12, because 12÷3=4 exactly

Multiple: A number you get by multiplying the original number by a whole number.

E.g. 12, 24, 36 are multiples of 12 (since 12×1, 12×2, 12×3, …)

Highest Common Factor (HCF): The largest factor that is common to two or more numbers.

Lowest Common Multiple (LCM): The smallest non-zero multiple common to two or more numbers.

1. Simplifying fractions


2. Sharing things equally


3. Scheduling repeating events (when two schedules meet again)


4. Packaging, splitting and many everyday problems

1. HCF
   
   
   1. Listing factors method
   
   
   2. Prime factorization method
   
   
   3. Short-division (division ladder / repeated division) method
   
   
2. LCM
   
   
   1. Listing multiples method
   
   
   2. Prime factorization method
   
   
   3. Using the relation (for two numbers): LCM(a,b) = (a × b) / HCF(a,b)

Here are many ways HCF & LCM show up in daily life, so you see why learning them is useful:

1. Splitting things into equal parts (no leftovers) – Use HCF. E.g. cutting cloth, sharing food, dividing students into equal groups.


2. Synchronising events that repeat at different intervals – Use LCM. E.g. two buses have different schedules – find when both arrive at the same time.


3. Scheduling maintenance / chores – If one task is every 10 days, another every 15 days, LCM gives you when both coincide.


4. Traffic light cycles – Lights changing at different intervals – LCM helps find when they all turn green together.


5. Music / Rhythm (beats) – For combining beats of different lengths, LCM finds when beats align in rhythm.


6. Baking / Cooking in batch sizes – If recipes need grouping of ingredients, HCF can help scale or divide without waste.


7. Packing and manufacturing – Boxes of different sizes, products in packs — LCM helps avoid leftovers.


8. Tiling a floor / mosaics – To tile floor with different tile sizes, you may need HCF to decide largest square tile that fits exactly.


9. Time management / event planning – When different tasks recur every x days, LCM helps find common deadlines.


10. Fractions operations (addition, subtraction) – To add fractions with different denominators, you use LCM of denominators (lowest common denominator).


11. Distributing gifts / share items equally – Use HCF to ensure each person gets the same number, without leftovers.


12. Repetitive patterns (clothes patterns, bead designs) – LCM helps in designing repeats of pattern so they align nicely.

HCF – Examples

1. 8 and 12 — Listing factors
   
   Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. Common: 1, 2, 4 → HCF = 4.


2. 15 and 20 — Prime factorization
   
   15 = 3 × 5. 20 = 2² × 5. Common prime: 5 → HCF = 5.


3. 18, 24 and 30 — Prime factorization
   
   18 = 2 × 3²; 24 = 2³ × 3; 30 = 2 × 3 × 5. Common primes: 2 and 3 with lowest powers 2¹ and 3¹ → HCF = 6.


4. 16 and 24 — Short division
   
   Divide by 2 repeatedly: (16,24) → (8,12) → (4,6) → (2,3) stop. Multiply divisors used together that divided both: 2×2×2 = 8.


5. 14 and 35 — Listing factors
   
   14: 1,2,7,14. 35: 1,5,7,35. Common: 7 → HCF = 7.


6. 21, 45, 105 — Prime factorization
   
   21 = 3 × 7; 45 = 3² × 5; 105 = 3 × 5 × 7. Common prime: 3 → HCF = 3.


7. 72 and 96 — Prime factorization
   
   72 = 2³ × 3²; 96 = 2⁵ × 3. Common: 2³ × 3 = 8 × 3 = 24.


8. 100 and 40 — Prime factorization
   
   100 = 2² × 5²; 40 = 2³ × 5. Common: 2² × 5 = 4 × 5 = 20.


9. 27, 36, 45 — Listing / prime
   
   27 = 3³; 36 = 2² × 3²; 45 = 3² × 5. Common: 3² = 9.


10. 48 and 180 — Prime factorization
    
    48 = 2⁴ × 3; 180 = 2² × 3² × 5. Common: 2² × 3 = 4 × 3 = 12.


11. HCF of 6 and 8

    Divide both by 2 → 3 and 4
    No more common divisor.
    HCF = 2
    

12. HCF of 12 and 16

    Divide both by 2 → 6 and 8
    Divide both by 2 → 3 and 4
    No more common divisor.
    HCF = 2 × 2 = 4
    

13. HCF of 15 and 25

    Divide both by 5 → 3 and 5
    No more common divisor.
    HCF = 5
    

14. HCF of 20 and 30

    Divide both by 2 → 10 and 15
    Divide both by 5 → 2 and 3
    HCF = 2 × 5 = 10
    

15. HCF of 18 and 24

    Divide both by 2 → 9 and 12
    Divide both by 3 → 3 and 4
    HCF = 2 × 3 = 6
    

16. HCF of 8 and 28

    Divide both by 2 → 4 and 14
    Divide both by 2 → 2 and 7
    HCF = 2 × 2 = 4
    

17. HCF of 9 and 27

    Divide both by 3 → 3 and 9
    Divide both by 3 → 1 and 3
    HCF = 3 × 3 = 9
    

18. HCF of 14 and 49

    Divide both by 7 → 2 and 7
    HCF = 7
    

19. HCF of 24 and 36

    Divide both by 2 → 12 and 18
    Divide both by 2 → 6 and 9
    Divide both by 3 → 2 and 3
    HCF = 2 × 2 × 3 = 12
    

20. HCF of 30 and 45

    Divide both by 3 → 10 and 15
    Divide both by 5 → 2 and 3
    HCF = 3 × 5 = 15
    

21. HCF of 32 and 40

    Divide both by 2 → 16 and 20
    Divide both by 2 → 8 and 10
    Divide both by 2 → 4 and 5
    HCF = 2 × 2 × 2 = 8
    

22. HCF of 35 and 50

    Divide both by 5 → 7 and 10
    HCF = 5
    

23. HCF of 25 and 30

    Divide both by 5 → 5 and 6
    HCF = 5
    

24. HCF of 21 and 63

    Divide both by 3 → 7 and 21
    Divide both by 7 → 1 and 3
    HCF = 3 × 7 = 21
    

25. HCF of 16 and 24

    Divide both by 2 → 8 and 12
    Divide both by 2 → 4 and 6
    Divide both by 2 → 2 and 3
    HCF = 2 × 2 × 2 = 8
    

26. HCF of 18 and 30

    Divide both by 2 → 9 and 15
    Divide both by 3 → 3 and 5
    HCF = 2 × 3 = 6
    

27. HCF of 40 and 60

    Divide both by 2 → 20 and 30
    Divide both by 2 → 10 and 15
    Divide both by 5 → 2 and 3
    HCF = 2 × 2 × 5 = 20
    

28. HCF of 27 and 36

    Divide both by 3 → 9 and 12
    Divide both by 3 → 3 and 4
    HCF = 3 × 3 = 9
    

29. HCF of 42 and 56

    Divide both by 2 → 21 and 28
    Divide both by 7 → 3 and 4
    HCF = 2 × 7 = 14
    

30. HCF of 12 and 18

    Divide both by 2 → 6 and 9
    Divide both by 3 → 2 and 3
    HCF = 2 × 3 = 6
    

1. 4 and 6 — Listing multiples
   
   Multiples of 4: 4, 8, 12, 16…; Multiples of 6: 6, 12, 18… → first common = 12.


2. 5 and 10 — Listing / prime
   
   Multiples of 5: 5,10,15…; Multiples of 10: 10,20… → LCM = 10.


3. 12 and 18 — Prime factorization
   
   12 = 2² × 3; 18 = 2 × 3². Take highest powers: 2² × 3² = 4 × 9 = 36.


4. 8, 9 and 12 — Prime factorization
   
   8 = 2³; 9 = 3²; 12 = 2² × 3. Highest powers: 2³ × 3² = 8 × 9 = 72.


5. 14 and 35 — Prime method / relation
   
   14 = 2 × 7; 35 = 5 × 7. LCM uses primes 2, 5, 7 → 2 × 5 × 7 = 70.


6. 21, 6 and 15 — Listing multiples / prime
   
   Multiples: 6 -> 6,12,18,24,30…; 15 -> 15,30…; 21 -> 21,42,63… → first common = 30.


7. 16 and 30 — Prime factorization
   
   16 = 2⁴; 30 = 2 × 3 × 5. Highest powers: 2⁴ × 3 × 5 = 16 × 3 × 5 = 240.


8. 18 and 24 — Prime method
   
   18 = 2 × 3²; 24 = 2³ × 3. Highest: 2³ × 3² = 8 × 9 = 72.


9. 40 and 90 — Prime method
   
   40 = 2³ × 5; 90 = 2 × 3² × 5. Highest: 2³ × 3² × 5 = 8 × 9 × 5 = 360.


10. 7, 11, 13 — Distinct primes
    
    7, 11 and 13 are distinct primes → LCM = 7 × 11 × 13 = 1001.


11. LCM of 2 and 3

    2 × 3 = 6
    LCM = 6
    

12. LCM of 4 and 5

    4 = 2 × 2
    5 is prime.
    Multiply all: 2 × 2 × 5 = 20
    LCM = 20
    

13. LCM of 3 and 6

    6 is already a multiple of 3.
    LCM = 6
    

14. LCM of 5 and 10

    10 is a multiple of 5.
    LCM = 10
    

15. LCM of 6 and 8

    Multiples of 6: 6,12,18,24
    Multiples of 8: 8,16,24
    First common = 24
    LCM = 24
    

16. LCM of 7 and 9

    Multiples of 7: 7,14,21,28,35,42,49,56,63
    Multiples of 9: 9,18,27,36,45,54,63
    First common = 63
    LCM = 63
    

17. LCM of 4 and 6

    Multiples of 4: 4,8,12,16
    Multiples of 6: 6,12,18
    First common = 12
    LCM = 12
    

18. LCM of 8 and 12

    Multiples of 8: 8,16,24
    Multiples of 12: 12,24,36
    First common = 24
    LCM = 24
    

19. LCM of 10 and 15

    Multiples of 10: 10,20,30
    Multiples of 15: 15,30,45
    First common = 30
    LCM = 30
    

20. LCM of 12 and 18

    Multiples of 12: 12,24,36
    Multiples of 18: 18,36,54
    First common = 36
    LCM = 36
    

21. LCM of 14 and 20

    Multiples of 14: 14,28,42,56,70
    Multiples of 20: 20,40,60,80
    First common = 140
    LCM = 140
    

22. LCM of 9 and 12

    Multiples of 9: 9,18,27,36
    Multiples of 12: 12,24,36,48
    First common = 36
    LCM = 36
    

23. LCM of 8 and 10

    Multiples of 8: 8,16,24,32,40
    Multiples of 10:10,20,30,40
    First common = 40
    LCM = 40
    

24. LCM of 4 and 9

    Multiples of 4: 4,8,12,16,20,24,28,32,36
    Multiples of 9: 9,18,27,36,45
    First common = 36
    LCM = 36
    

25. LCM of 5 and 12

    Multiples of 5: 5,10,15,20,25,30,35,40,45,50,55,60
    Multiples of 12:12,24,36,48,60
    First common = 60
    LCM = 60
    

26. LCM of 6 and 15

    Multiples of 6: 6,12,18,24,30
    Multiples of 15:15,30,45
    First common = 30
    LCM = 30
    

27. LCM of 7 and 8

    Multiples of 7: 7,14,21,28,35,42,49,56
    Multiples of 8: 8,16,24,32,40,48,56
    First common = 56
    LCM = 56
    

28. LCM of 11 and 13

    11 and 13 are prime.
    LCM = 11 × 13 = 143
    

29. LCM of 3, 4 and 5

    3 × 4 × 5 = 60 (because none is multiple of another)
    LCM = 60
    

30. LCM of 4, 6 and 8

    Multiples of 8: 8,16,24
    Check 24 is multiple of 6 and 4.
    LCM = 24
    

1. Find HCF and LCM of 9 and 12.


2. Find HCF and LCM of 15 and 25.


3. Find HCF and LCM of 18 and 30.


4. Find HCF and LCM of 8, 12, 20.


5. Find HCF and LCM of 14, 21, 28.


6. Find HCF and LCM of 24, 36, 60.


7. Find HCF and LCM of 32 and 48.


8. Find HCF and LCM of 45, 75, 90.


9. Find HCF and LCM of 27, 72, 108.


10. Find HCF and LCM of 100, 150, 250.

Worked Answers (brief)

1. 9 & 12 → HCF = 3 ; LCM = 36


2. 15 & 25 → HCF = 5 ; LCM = 75


3. 18 & 30 → HCF = 6 ; LCM = 90


4. 8,12,20 → HCF = 4 ; LCM = 120


5. 14,21,28 → HCF = 7 ; LCM = 84


6. 24,36,60 → HCF = 12 ; LCM = 360


7. 32 & 48 → HCF = 16 ; LCM = 96


8. 45,75,90 → HCF = 15 ; LCM = 450


9. 27,72,108 → HCF = 9 ; LCM = 216


10. 100,150,250 → HCF = 50 ; LCM = 750

1. Confusing “largest common factor” with the largest of the original numbers.


2. Using highest power for HCF instead of lowest power (prime method mistake).


3. Using lowest power for LCM instead of highest power (prime method mistake).


4. Trying pairwise LCM/HCF steps incorrectly for more than two numbers — prefer prime factor method for >2 numbers.

Teaching Tips

1. Start pupils on listing methods to build intuition, then move to prime factorization for speed and reliability.


2. Use real-life examples: class grouping, bell schedules, fraction addition.


3. Practice prime factor trees until pupils are fluent.

Extra Practice (challenge)

1. Find HCF and LCM of 126 and 210.


2. Three friends meet every 8, 12 and 20 days. If they met today, after how many days will they next meet together?


3. Find the greatest length (cm) to cut a 630 cm and 945 cm mat into equal pieces with none left over (use HCF).


4. Find the least time when events repeating every 45 min and 90 min will coincide (use LCM).