Class 8 Mathematics — Exploring Multiples and Factors

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: You want to arrange 24 students in equal rows for a school photo. The possible row sizes are factors of 24: 1, 2, 3, 4, 6, 8, 12, or 24. Traffic lights at two different junctions blink at intervals of 12 and 18 seconds — they will next blink together after LCM(12, 18) = 36 seconds. HCF and LCM solve everyday problems from timetables to tiling floors!

🔢 Topic 1: What Are Multiples?

Multiples come from skip counting.
Multiply a number by 1, 2, 3, 4 ...
Multiples of 3: 3, 6, 9, 12, 15 ...
Multiples of 5: 5, 10, 15, 20, 25 ...
Every number has endless multiples.
The smallest multiple is the number itself.

🔢 Topic 2: What Are Factors?

A factor divides a number evenly.
There is no remainder left over.
Factors of 12: 1, 2, 3, 4, 6, 12.
Factors of 20: 1, 2, 4, 5, 10, 20.
Every number has a limited set of factors.
1 is a factor of every number.
The number itself is always a factor.

🔢 Topic 3: Prime and Composite Numbers

A prime number has exactly 2 factors.
Its factors are 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17 ...
2 is the only even prime number.
A composite number has more than 2 factors.
Examples: 4, 6, 8, 9, 10, 12, 14 ...
The number 1 is neither prime nor composite.

🔢 Topic 4: Prime Factorisation

Break a number into prime factors.
Use a factor tree to split it.
Keep splitting until all are prime.
Example: 36 = 2 × 2 × 3 × 3
We write this as 36 = 2² × 3².
Every composite number has a unique set.
Only the order of primes can change.

Every composite number = product of primes

🔢 Topic 5: Highest Common Factor (HCF)

HCF = Highest Common Factor.
Also called GCD (Greatest Common Divisor).
Find prime factors of each number.
Pick the smallest power of shared primes.
Multiply those together = HCF.
Example: HCF of 12 and 18.
12 = 2² × 3, and 18 = 2 × 3².
Shared: 2¹ and 3¹. HCF = 6.

🔢 Topic 6: Lowest Common Multiple (LCM)

LCM = Lowest Common Multiple.
The smallest number both can divide.
Find prime factors of each number.
Pick the greatest power of all primes.
Multiply those together = LCM.
Example: LCM of 12 and 18.
Greatest powers: 2² and 3².
LCM = 4 × 9 = 36.

🔢 Topic 7: HCF × LCM = Product Rule

For any two numbers a and b:
HCF(a, b) × LCM(a, b) = a × b
Check: HCF(12, 18) = 6, LCM = 36.
6 × 36 = 216.
12 × 18 = 216. They match!
This rule works for two numbers only.

HCF(a, b) × LCM(a, b) = a × b

🔢 Topic 8: Co-Prime Numbers

Two numbers are co-prime if HCF = 1.
They share no common factor except 1.
Example: 8 and 15 are co-prime.
Factors of 8: 1, 2, 4, 8.
Factors of 15: 1, 3, 5, 15.
Only 1 is shared. So HCF = 1.
Co-prime does not mean both are prime.

🔢 Topic 9: Divisibility Rules Recap

By 2: last digit is 0, 2, 4, 6, or 8.
By 3: sum of digits divides by 3.
By 4: last two digits divide by 4.
By 5: last digit is 0 or 5.
By 6: divisible by both 2 and 3.
By 8: last three digits divide by 8.
By 9: sum of digits divides by 9.
By 10: last digit is 0.

🔢 Topic 10: Fun Patterns in Multiples

Multiples of 9: digits always add to 9.
9, 18, 27, 36 → 9, 9, 9, 9!
Multiples of 5 always end in 0 or 5.
Multiples of 11: alternating digit sums match.
Example: 121 → 1−2+1 = 0 (divisible!).
Even × Even = Even.
Odd × Odd = Odd.
Even × Odd = Even.

🎨 Section 2 — Easy Illustrations

Picture 1: Factor Tree for 180

Click "Play" to see the factor tree grow step by step. Green circles = prime factors.

Picture 2: Venn Diagram — HCF and LCM of 24 and 36

HCF = shared factors. LCM = all factors with highest powers.

Picture 3: Sieve of Eratosthenes (Primes to 50)

Coloured boxes are prime. Light boxes are composite. 1 is grey (neither).

Picture 4: Divisibility Rules Chart

Use these quick checks to test if a number divides evenly.

Picture 5: Co-Prime Number Pairs

Co-prime means the only shared factor is 1. The numbers themselves can be composite.

✏️ Section 3 — Worked Examples

Example 1: Find HCF and LCM of 28 and 42

Question: Find HCF and LCM of 28 and 42 using prime factorisation.

1 Write 28 as primes: 28 = 2 × 2 × 7 = 2² × 7.

2 Write 42 as primes: 42 = 2 × 3 × 7 = 2 × 3 × 7.

3 HCF = smallest powers of shared primes.

4 Shared primes: 2 and 7. Smallest: 2¹ and 7¹.

5 HCF = 2 × 7 = 14.

6 LCM = greatest powers of all primes.

7 All primes: 2², 3¹, 7¹. LCM = 4 × 3 × 7 = 84.

HCF = 14, LCM = 84

Check: 14 × 84 = 1176 = 28 × 42 ✓

Example 2: Verify HCF × LCM = Product of 15 and 20

Question: Check that HCF × LCM = 15 × 20.

1 Write 15 as primes: 15 = 3 × 5.

2 Write 20 as primes: 20 = 2² × 5.

3 Shared prime: 5. Smallest power: 5¹.

4 HCF = 5.

5 All primes: 2², 3¹, 5¹. LCM = 4 × 3 × 5 = 60.

6 HCF × LCM = 5 × 60 = 300.

7 15 × 20 = 300. Both sides match!

5 × 60 = 300 = 15 × 20 ✓ Verified!

Rule: HCF(a,b) × LCM(a,b) = a × b always works!

Example 3: When Do Three Bells Ring Together?

Question: Three bells ring every 6, 8, and 12 minutes. When will they all ring together next?

1 We need the LCM of 6, 8, and 12.

2 Write 6 as primes: 6 = 2 × 3.

3 Write 8 as primes: 8 = 2³.

4 Write 12 as primes: 12 = 2² × 3.

5 Greatest powers: 2³ and 3¹.

6 LCM = 8 × 3 = 24.

All three bells ring together after 24 minutes!

LCM finds the first time events repeat together.

Term	What It Means	Key Fact
Multiple	Skip counting result	3, 6, 9, 12 ...
Factor	Divides a number evenly	No remainder
Prime Number	Exactly 2 factors	2, 3, 5, 7, 11...
Composite	More than 2 factors	4, 6, 8, 9...
Prime Factorisation	Break into prime factors	Use a factor tree
HCF	Highest Common Factor	Smallest shared powers
LCM	Lowest Common Multiple	Greatest all powers
HCF × LCM Rule	HCF × LCM = a × b	Two numbers only
Co-Prime	HCF is 1	8 & 15 are co-prime
Divisibility	Quick check rules	By 2, 3, 4, 5, 6...

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. Multiples of 4 are: 4, 8, _____, 16, _____.
2. HCF stands for _________ Common _________.
3. The number 1 is neither _________ nor _________.
4. Two numbers with HCF = 1 are called _________.
5. A number divisible by 2 and 3 is also divisible by _____.

✓ B. True or False

1. Every prime number is odd. _______
2. HCF of two numbers is always smaller than both. _______
3. HCF(a,b) × LCM(a,b) = a × b for any two numbers. _______
4. 15 and 28 are co-prime numbers. _______
5. Every number is a multiple of itself. _______

✏️ C. Solve These

1. Find the HCF and LCM of 36 and 48.

Show your work here

2. Three traffic lights blink every 4, 6, and 8 seconds. When will they blink together?

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. HCF(8, 15)	a. 60
2. LCM(12, 20)	b. Composite
3. The number 27	c. 1 (co-prime)
4. Factors of 6	d. 1, 2, 3, 6

🎨 E. Draw the Factor Tree

Draw the factor tree for 120 and write its prime factorisation.

Draw your factor tree here

✅ Detailed Solutions — Section 5 Handout

A. Fill in the Blanks — Answers

1. Multiples of 4 are: 4, 8, 12, 16, 20.

2. HCF stands for Highest Common Factor.

3. The number 1 is neither prime nor composite.

4. Two numbers with HCF = 1 are called co-prime.

5. A number divisible by 2 and 3 is also divisible by 6. (2 × 3 = 6, and they share no common factor.)

B. True or False — Answers

1. Every prime number is odd. — False. 2 is a prime number and it is even.

2. HCF of two numbers is always smaller than both. — False. If one divides the other (e.g., 6 and 12), HCF = 6 = one of the numbers.

3. HCF(a,b) × LCM(a,b) = a × b for any two numbers. — True. This is a key property. ✓

4. 15 and 28 are co-prime numbers. — True. 15 = 3 × 5, 28 = 2² × 7. No common factor except 1. ✓

5. Every number is a multiple of itself. — True. n × 1 = n. ✓

C1. Find the HCF and LCM of 36 and 48

1 Prime factorisation of 36: 36 = 2 × 18 = 2 × 2 × 9 = 2² × 3².

2 Prime factorisation of 48: 48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2⁴ × 3.

3 HCF = product of lowest powers of common primes = 2² × 3¹ = 4 × 3 = 12.

4 LCM = product of highest powers of all primes = 2⁴ × 3² = 16 × 9 = 144.

5 Check: HCF × LCM = 12 × 144 = 1728 = 36 × 48. ✓

HCF = 12, LCM = 144

C2. Traffic lights blink every 4, 6, and 8 seconds. When will they blink together?

1 They blink together again after LCM(4, 6, 8) seconds.

2 4 = 2², 6 = 2 × 3, 8 = 2³.

3 LCM = highest power of each prime = 2³ × 3 = 8 × 3 = 24.

They will blink together after 24 seconds.

D. Match the Columns — Answers

1. HCF(8, 15) → c. 1 (co-prime). (8 = 2³, 15 = 3 × 5. No common factors.)

2. LCM(12, 20) → a. 60. (12 = 2²×3, 20 = 2²×5. LCM = 2²×3×5 = 60.)

3. The number 27 → b. Composite. (27 = 3 × 3 × 3. Has factors other than 1 and itself.)

4. Factors of 6 → d. 1, 2, 3, 6.

🃏 Section 6 — Revision Flashcards

👉

🔢

What is a multiple?

A multiple is the result of skip counting. Multiply the number by 1, 2, 3, 4 ...

Example: multiples of 7 are 7, 14, 21, 28 ...

🔎

What is a factor?

A factor is a number that divides evenly with no remainder.

Factors of 12: 1, 2, 3, 4, 6, 12

⭐

What is a prime number?

A number with exactly 2 factors: 1 and itself.

Examples: 2, 3, 5, 7, 11, 13 ...

🌳

What is prime factorisation?

Breaking a number into a product of prime factors using a factor tree.

36 = 2² × 3²

📈

How do you find HCF?

Write both numbers as primes. Pick the smallest power of each shared prime. Multiply them.

📊

How do you find LCM?

Write both numbers as primes. Pick the greatest power of every prime. Multiply them all.

🔗

What is the HCF × LCM rule?

HCF(a,b) × LCM(a,b) = a × b

Works for two numbers only.

🤝

What are co-prime numbers?

Two numbers whose HCF is 1. They share no common factor except 1.

Example: 8 and 15 are co-prime.