Notes For All Chapters – Maths Class 6 Ganita Prakash
📘 Class 6 · Mathematics · Ganita Prakash
⏱️ Prime Time
Chapter 5 — Factors, Multiples, Prime & Composite Numbers, Prime Factorisation and Divisibility Tests
Common Multiples
Prime Numbers
Co-prime Numbers
Prime Factorisation
Divisibility Tests
Prime Numbers
Co-prime Numbers
Prime Factorisation
Divisibility Tests
📋 Contents
5.1 Common Multiples and Common Factors
Idli-Vada Game
Children sit in a circle and count from 1. When a number is a multiple of 3, they say “Idli”. When it is a multiple of 5, they say “Vada”. When it is a multiple of both 3 and 5, they say “Idli-Vada”! If a player makes a mistake, they are out.
Children sit in a circle and count from 1. When a number is a multiple of 3, they say “Idli”. When it is a multiple of 5, they say “Vada”. When it is a multiple of both 3 and 5, they say “Idli-Vada”! If a player makes a mistake, they are out.
🔢 Multiples
A multiple of a number is obtained by multiplying it with natural numbers (1, 2, 3, …).
Multiples of 3
3, 6, 9, 12, 15, 18, 21, 24, 27, 30 …
3, 6, 9, 12, 15, 18, 21, 24, 27, 30 …
Multiples of 5
5, 10, 15, 20, 25, 30, 35, 40, 45, 50 …
5, 10, 15, 20, 25, 30, 35, 40, 45, 50 …
🔗 Common Multiples
Numbers that appear in the multiples of both numbers are called Common Multiples.
Common multiples of 3 and 5 = 15, 30, 45, 60, 75 …
Remember!
The smallest common multiple of 3 and 5 is 15. These are called common multiples. In the Idli-Vada game, “Idli-Vada” is said for these numbers.
The smallest common multiple of 3 and 5 is 15. These are called common multiples. In the Idli-Vada game, “Idli-Vada” is said for these numbers.
📦 Factors (Divisors)
In the Jump Jackpot game, Jumpy jumps in equal steps starting from 0. The numbers that divide a given number exactly (with remainder 0) are called its factors or divisors.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
- Every number is a factor of itself.
- 1 is a factor of every number.
- The smallest factor of any number is 1.
- The largest factor of a number is the number itself.
🤝 Common Factors
Factors shared by two or more numbers are called their common factors.
Factors of 14 = 1, 2, 7, 14
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Common Factors of 14 and 36 = 1 and 2
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Common Factors of 14 and 36 = 1 and 2
Exam Tip
In the Jump Jackpot game, the jump sizes that land on BOTH treasure numbers are the common factors of those two numbers.
In the Jump Jackpot game, the jump sizes that land on BOTH treasure numbers are the common factors of those two numbers.
✅ Example: Find Common Factors
| Numbers | Factors of First | Factors of Second | Common Factors |
|---|---|---|---|
| 20 and 28 | 1,2,4,5,10,20 | 1,2,4,7,14,28 | 1, 2, 4 |
| 35 and 50 | 1,5,7,35 | 1,2,5,10,25,50 | 1, 5 |
| 4, 8 and 12 | 1,2,4 | 1,2,4,8 | 1, 2, 4 |
| 5, 15 and 25 | 1,5 | 1,3,5,15 | 1, 5 |
🧩
Perfect Numbers! A number is called a perfect number if the sum of all its factors equals twice the number. Example: 28 → factors: 1+2+4+7+14+28 = 56 = 2×28. The perfect number between 1 and 10 is 6 (1+2+3+6 = 12 = 2×6).
5.2 Prime Numbers
📐 Rectangular Arrangement Idea
Guna packs 12 figs in many rectangular ways (1×12, 2×6, 3×4, 4×3…), but Anshu packing 7 figs can only do it one way: 1×7. This is because 12 has many factors, but 7 has only 2 factors — 1 and 7!
🟢 Prime Numbers
A number with exactly 2 factors (1 and itself) is called a Prime Number.
First few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 …
Key Fact
2 is the smallest prime number and the only even prime. All other prime numbers are odd.
2 is the smallest prime number and the only even prime. All other prime numbers are odd.
🔴 Composite Numbers
A number with more than 2 factors is called a Composite Number.
First few composites: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 …
⚠️ The Special Case: 1
Important!
The number 1 is neither prime nor composite. It has only one factor — itself.
The number 1 is neither prime nor composite. It has only one factor — itself.
🧪 Sieve of Eratosthenes
An ancient Greek mathematician named Eratosthenes (about 2200 years ago) gave a method to find all prime numbers.
- Cross out 1 — it is neither prime nor composite.
- Circle 2, then cross out all multiples of 2 (4, 6, 8, …).
- Circle 3 (next uncrossed), then cross out all multiples of 3 (6, 9, 12, …).
- Circle 5 (next uncrossed), then cross out all multiples of 5 (10, 15, 20, …).
- Continue until all numbers are either circled or crossed out.
- All circled numbers = Prime Numbers. All crossed numbers (except 1) = Composite Numbers.
Twin Primes
Pairs of prime numbers with a difference of 2 are called Twin Primes. Examples: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73).
Pairs of prime numbers with a difference of 2 are called Twin Primes. Examples: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73).
🌟
Amazing Fact! The largest known prime number is so huge that it would take around 6500 pages to write it down — it can only be stored on a computer!
📊 Primes Between 1–50 (Quick Reference)
| Range | Prime Numbers | Count |
|---|---|---|
| 1–10 | 2, 3, 5, 7 | 4 |
| 11–20 | 11, 13, 17, 19 | 4 |
| 21–30 | 23, 29 | 2 |
| 31–40 | 31, 37 | 2 |
| 41–50 | 41, 43, 47 | 3 |
5.3 Co-prime Numbers
🎮 Safe Pairs in Jump Jackpot
In the upgraded treasure game, Jumpy cannot use a jump size of 1. Grumpy hides treasures on two numbers. A pair is “safe” only if Jumpy cannot reach both — meaning the two numbers have no common factor other than 1.
Two numbers are Co-prime if their only common factor is 1.
Note
Co-prime numbers do NOT need to be prime themselves! For example, 4 and 9 are co-prime (their only common factor is 1), but neither 4 nor 9 is a prime number.
Co-prime numbers do NOT need to be prime themselves! For example, 4 and 9 are co-prime (their only common factor is 1), but neither 4 nor 9 is a prime number.
✅ Examples
| Pair | Common Factors | Co-prime? |
|---|---|---|
| 4 and 9 | 1 only | ✅ Yes |
| 15 and 39 | 1, 3 | ❌ No (have 3) |
| 18 and 35 | 1 only | ✅ Yes |
| 81 and 18 | 1, 3, 9 | ❌ No |
| 15 and 37 | 1 only | ✅ Yes |
Co-prime and LCM Connection
When two numbers are co-prime, their first common multiple (LCM) equals their product. For example, 4 × 9 = 36, and the LCM of 4 and 9 is 36. When they are NOT co-prime, LCM is less than the product.
When two numbers are co-prime, their first common multiple (LCM) equals their product. For example, 4 × 9 = 36, and the LCM of 4 and 9 is 36. When they are NOT co-prime, LCM is less than the product.
5.4 Prime Factorisation
📖 What is Prime Factorisation?
Writing a number as a product of prime numbers only is called its Prime Factorisation. The prime numbers in this product are called prime factors.
Every number greater than 1 has a unique prime factorisation (only the order may differ).
🔎 Step-by-Step: Prime Factorisation of 56
56 = 4 × 14
= 4 × 2 × 7
= 2 × 2 × 2 × 7
Prime factorisation: 56 = 2 × 2 × 2 × 7
= 4 × 2 × 7
= 2 × 2 × 2 × 7
Prime factorisation: 56 = 2 × 2 × 2 × 7
🌲 More Examples
36 = ?
36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3
36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3
72 = ?
72 = 12 × 6 = (2×2×3) × (2×3) = 2 × 2 × 2 × 3 × 3
72 = 12 × 6 = (2×2×3) × (2×3) = 2 × 2 × 2 × 3 × 3
63 = ?
63 = 9 × 7 = 3 × 3 × 7 = 3 × 3 × 7
63 = 9 × 7 = 3 × 3 × 7 = 3 × 3 × 7
84 = ?
84 = 4 × 21 = 2 × 2 × 3 × 7 = 2 × 2 × 3 × 7
84 = 4 × 21 = 2 × 2 × 3 × 7 = 2 × 2 × 3 × 7
Key Rule
The number 1 has no prime factorisation — it is not divisible by any prime. A prime number’s only prime factorisation is itself (e.g., 7 = 7).
The number 1 has no prime factorisation — it is not divisible by any prime. A prime number’s only prime factorisation is itself (e.g., 7 = 7).
🤝 Using Prime Factorisation to Check Co-primeness
Find the prime factorisation of both numbers. If they share no common prime factor, they are co-prime.
56 = 2 × 2 × 2 × 7
63 = 3 × 3 × 7
Common prime factor: 7 → NOT co-prime!80 = 2 × 2 × 2 × 2 × 5
63 = 3 × 3 × 7
No common prime factor → Co-prime! ✅
63 = 3 × 3 × 7
Common prime factor: 7 → NOT co-prime!80 = 2 × 2 × 2 × 2 × 5
63 = 3 × 3 × 7
No common prime factor → Co-prime! ✅
➗ Checking Divisibility Using Prime Factorisation
A number A is divisible by B if the prime factorisation of B is completely included in the prime factorisation of A.
Is 168 divisible by 12?
168 = 2 × 2 × 2 × 3 × 7
12 = 2 × 2 × 3 (all these appear in 168!)
Yes! 168 ÷ 12 = 14 ✅Is 75 divisible by 21?
75 = 3 × 5 × 5
21 = 3 × 7 (7 does NOT appear in 75!)
No! ❌
168 = 2 × 2 × 2 × 3 × 7
12 = 2 × 2 × 3 (all these appear in 168!)
Yes! 168 ÷ 12 = 14 ✅Is 75 divisible by 21?
75 = 3 × 5 × 5
21 = 3 × 7 (7 does NOT appear in 75!)
No! ❌
5.5 Divisibility Tests
We can check divisibility of large numbers without doing long division by using simple rules based on the digits.
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 (even) | 8560 ✅ (ends in 0) |
| 4 | Last two digits form a number divisible by 4 | 8536 → 36÷4=9 ✅ |
| 5 | Last digit is 0 or 5 | 8560 ✅; 8565 ✅ |
| 8 | Last three digits form a number divisible by 8 | 8560 → 560÷8=70 ✅ |
| 10 | Last digit is 0 | 8560 ✅; 8563 ❌ |
Coming in Higher Classes
Divisibility rules for 3, 6, 7, and 9 will be studied in later classes. (Hint: for 3 and 9, the sum of digits is used!)
Divisibility rules for 3, 6, 7, and 9 will be studied in later classes. (Hint: for 3 and 9, the sum of digits is used!)
🔗 Relationship Between Divisibility Rules
Smart Trick!
If a number is divisible by both 2 and 5, it is divisible by 10. Since 10 = 2 × 5, and 2 and 5 are co-prime, Guna could check just 2 and 5 to confirm divisibility by 2, 4, 5, 8, and 10 for 14560.
If a number is divisible by both 2 and 5, it is divisible by 10. Since 10 = 2 × 5, and 2 and 5 are co-prime, Guna could check just 2 and 5 to confirm divisibility by 2, 4, 5, 8, and 10 for 14560.
📅 Leap Year Rule
- A year is a leap year if it is divisible by 4.
- Exception: Years divisible by 100 are NOT leap years…
- Exception to exception: …unless also divisible by 400, in which case they ARE leap years.
- Examples: 2024 ✅ leap; 1900 ❌ not leap (divisible by 100 but not 400); 2000 ✅ leap (divisible by 400).
Important Questions & Answers
Q1. In the Idli-Vada game (multiples of 3 = Idli, multiples of 5 = Vada), at what number is “Idli-Vada” said for the 10th time?
Ans: “Idli-Vada” is said at multiples of both 3 and 5, i.e., multiples of 15. The 10th such number is 15 × 10 = 150.
Q2. Find the prime factorisation of 360.
Ans: 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2 × 2 × 2 × 9 × 5 = 2³ × 3² × 5 = 2 × 2 × 2 × 3 × 3 × 5
Q3. Are 18 and 35 co-prime? Verify using prime factorisation.
Ans: 18 = 2 × 3 × 3 and 35 = 5 × 7. There are no common prime factors. Therefore, 18 and 35 are co-prime ✅.
Q4. Is 168 divisible by 24? Use prime factorisation.
Ans: 168 = 2 × 2 × 2 × 3 × 7 and 24 = 2 × 2 × 2 × 3. The prime factorisation of 24 is fully included in 168. So 168 ÷ 24 = 7 ✅.
Q5. Find the smallest number that is a multiple of all numbers from 1 to 10.
Ans: We need LCM of 1–10. Prime factorisations: 2, 3, 4=2², 5, 6=2×3, 7, 8=2³, 9=3², 10=2×5. LCM = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520.
Q6. Which of the following numbers are prime? 23, 51, 37, 26
Ans: 23 (only factors: 1 and 23) and 37 (only factors: 1 and 37) are prime. 51 = 3 × 17 (composite); 26 = 2 × 13 (composite).
Q7. I am a number less than 40. One of my factors is 7. The sum of my digits is 8. Who am I?
Ans: Multiples of 7 less than 40: 7, 14, 21, 28, 35. Sum of digits: 7(7), 5(14), 3(21), 10(28), 8(35✅). Answer = 35.
Q8. In the treasure game, treasures are on 28 and 70. What jump sizes land on both?
Ans: We need common factors of 28 and 70. 28 = 2² × 7 and 70 = 2 × 5 × 7. Common factors = 1, 2, 7, 14. Jump sizes = 1, 2, 7, 14.
Quick Summary — All Key Concepts
🔢 Factor
If A ÷ B = exact whole number, then B is a factor of A. Every number has at least 2 factors: 1 and itself.
If A ÷ B = exact whole number, then B is a factor of A. Every number has at least 2 factors: 1 and itself.
📐 Multiple
A multiple of a number is obtained by multiplying it with 1, 2, 3 … Every number is a multiple of 1 and itself.
A multiple of a number is obtained by multiplying it with 1, 2, 3 … Every number is a multiple of 1 and itself.
🟢 Prime Number
Has exactly 2 factors (1 and itself). e.g., 2, 3, 5, 7, 11 … 2 is the only even prime.
Has exactly 2 factors (1 and itself). e.g., 2, 3, 5, 7, 11 … 2 is the only even prime.
🔴 Composite Number
Has more than 2 factors. e.g., 4, 6, 8, 9 … 1 is neither prime nor composite.
Has more than 2 factors. e.g., 4, 6, 8, 9 … 1 is neither prime nor composite.
🤝 Co-prime Numbers
Two numbers with only 1 as common factor. Not required to be prime. e.g., 4 and 9.
Two numbers with only 1 as common factor. Not required to be prime. e.g., 4 and 9.
🌳 Prime Factorisation
Writing a number as a product of primes only. It is unique (except for order). e.g., 36 = 2×2×3×3.
Writing a number as a product of primes only. It is unique (except for order). e.g., 36 = 2×2×3×3.
⚙️ Sieve of Eratosthenes
Ancient method to find all primes by crossing out multiples. Developed ~2200 years ago.
Ancient method to find all primes by crossing out multiples. Developed ~2200 years ago.
🧪 Divisibility by 2
Last digit is even (0,2,4,6,8). e.g., 8560 ✅
Last digit is even (0,2,4,6,8). e.g., 8560 ✅
🧪 Divisibility by 5
Last digit is 0 or 5. e.g., 8565 ✅
Last digit is 0 or 5. e.g., 8565 ✅
🧪 Divisibility by 4
Last two digits divisible by 4. e.g., 8536 → 36÷4=9 ✅
Last two digits divisible by 4. e.g., 8536 → 36÷4=9 ✅
🧪 Divisibility by 8
Last three digits divisible by 8. e.g., 8560 → 560÷8=70 ✅
Last three digits divisible by 8. e.g., 8560 → 560÷8=70 ✅
🧪 Divisibility by 10
Last digit is 0. e.g., 8560 ✅, 8563 ❌
Last digit is 0. e.g., 8560 ✅, 8563 ❌
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