🌟 1. FACTORS
🔹 What exactly is a factor?
A factor is a number that can divide another number completely (remainder = 0).
👉 Think like this:
If a number can be made by multiplying two numbers, then those numbers are its factors.
✅ Example
12 = 3 × 4
So, 3 and 4 are factors of 12
🔹 How to find factors?
👉 Divide the number by smaller numbers:
For 12:
- 12 ÷ 1 = 12 ✔
- 12 ÷ 2 = 6 ✔
- 12 ÷ 3 = 4 ✔
- 12 ÷ 4 = 3 ✔
- 12 ÷ 5 = ❌
👉 So factors = 1, 2, 3, 4, 6, 12
🔹 Important Understanding
- Factors always come in pairs
Example:
12 → (1,12), (2,6), (3,4) - You only check till √number (for higher classes later)
🔹 Why factors matter?
- Used in HCF
- Helps identify prime/composite numbers
🌟 2. MULTIPLES
🔹 What are multiples?
Multiples are numbers you get when you keep adding or multiplying a number
👉 Think:
Multiples of 5 = 5×1, 5×2, 5×3…
✅ Example
Multiples of 4 →
4, 8, 12, 16, 20…
🔹 Key Idea
- Factors go inside the number
- Multiples go outside the number
👉 Example:
- Factors of 6 → 1,2,3,6
- Multiples of 6 → 6,12,18,24…
🔹 Why multiples matter?
- Used in LCM
- Used in real life (like time cycles, patterns)
🌟 3. PRIME & COMPOSITE NUMBERS
🔹 Prime Numbers (Very Important Concept)
A number that has only 2 factors:
👉 1 and itself
✅ Example: 7
- Factors → 1, 7
👉 Only 2 → So PRIME
🔹 Why is 2 special?
- Only even number with 2 factors
👉 All other even numbers are divisible by 2 → so not prime
🔹 Composite Numbers
Numbers with more than 2 factors
✅ Example: 8
Factors → 1, 2, 4, 8
👉 More than 2 → COMPOSITE
🔹 Special Case: 1
- Only 1 factor
👉 So:
❌ Not prime
❌ Not composite
🔹 Real Understanding Trick
- Prime = “cannot be broken”
- Composite = “can be broken into smaller parts”
🌟 4. TESTS OF DIVISIBILITY (LOGIC)
Instead of dividing, we use short tricks
🔹 Divisible by 2
👉 Last digit even
Example: 246 ✔
🔹 Divisible by 3
👉 Add digits
Example: 123
1+2+3 = 6 → divisible by 3 ✔
🔹 Divisible by 4
👉 Last 2 digits
Example: 316 → 16 ÷ 4 ✔
🔹 Divisible by 5
👉 Ends with 0 or 5
🔹 Divisible by 6
👉 Must pass BOTH:
- divisible by 2
- divisible by 3
🔹 Divisible by 9
👉 Sum of digits divisible by 9
🔹 Why these rules?
They come from how numbers are built in place value system (base 10).
🌟 5. PRIME FACTORISATION
🔹 What is it?
Breaking a number into only prime numbers
✅ Example: 24
24 = 2 × 12
= 2 × 2 × 6
= 2 × 2 × 2 × 3
👉 Final = 2³ × 3
🔹 Why only primes?
Because primes are the basic building blocks of numbers
👉 Like atoms in chemistry
🔹 Methods
- Factor Tree
- Division Method
🌟 6. HCF (Highest Common Factor)
🔹 Meaning
The biggest number that divides all given numbers
✅ Example: 12 and 18
Factors:
- 12 → 1,2,3,4,6,12
- 18 → 1,2,3,6,9,18
👉 Common = 1,2,3,6
👉 Highest = 6
🔹 Using Prime Factorisation
12 = 2² × 3
18 = 2 × 3²
👉 Common = 2 × 3 = 6
🔹 Real-Life Use
- Dividing things equally
- Grouping objects
🌟 7. LCM (Lowest Common Multiple)
🔹 Meaning
The smallest number that both numbers can reach
✅ Example: 4 and 6
Multiples:
- 4 → 4,8,12…
- 6 → 6,12…
👉 First common = 12
🔹 Using Prime Factorisation
4 = 2²
6 = 2 × 3
👉 Take highest powers
👉 LCM = 2² × 3 = 12
🔹 Real-Life Use
- Time cycles (bells ringing together)
- Repeating events
🌟 8. RELATION BETWEEN HCF & LCM
👉 For 2 numbers:
HCF × LCM = Product of numbers
✅ Example
Numbers = 6 and 8
- HCF = 2
- LCM = 24
👉 2 × 24 = 48 = 6 × 8 ✔
🔹 Why important?
Helps solve faster in exams
🌟 9. EVEN & ODD NUMBERS
🔹 Even Numbers
Divisible by 2
👉 End in: 0,2,4,6,8
🔹 Odd Numbers
Not divisible by 2
👉 End in: 1,3,5,7,9
🔹 Pattern Logic
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd
👉 Helps in quick solving
🌟 10. CO-PRIME NUMBERS
🔹 Meaning
Numbers having only 1 as common factor
✅ Example
8 → factors: 1,2,4,8
15 → factors: 1,3,5,15
👉 Common = only 1
👉 So co-prime
🔹 Important Point
- Co-prime numbers don’t need to be prime
Example:
8 and 15 → co-prime ✔
🧠 FINAL UNDERSTANDING (BIG IDEA)
- Numbers are made of prime building blocks
- HCF → what is common
- LCM → what is shared in future multiples
- Divisibility rules → shortcuts
🎯 PRO TIP (VERY IMPORTANT)
👉 If you truly understand:
- Factors
- Multiples
- Prime factorisation
Then:
👉 HCF & LCM become VERY EASY