Playing with Numbers — Class 6 Mathematics
'Every number tells a story — factors, multiples, and primes are its characters.'
1. Introduction
Numbers are not just counting tools — they have relationships! Factors divide a number exactly. Multiples are numbers you get by multiplying. Prime numbers are the 'atoms' of mathematics — they cannot be broken into smaller factors.
Why This Chapter Matters
- HCF and LCM are used in fractions, ratios, and real-life problems
- Divisibility tests save time in exams
- Prime factorisation is the foundation of higher number theory
2. Factors and Multiples
Factors
A factor of a number divides it exactly (remainder = 0).
Example: Factors of 12: 1, 2, 3, 4, 6, 12
- 12 ÷ 1 = 12 ✓
- 12 ÷ 2 = 6 ✓
- 12 ÷ 3 = 4 ✓
- 12 ÷ 4 = 3 ✓
- 12 ÷ 6 = 2 ✓
- 12 ÷ 12 = 1 ✓
Properties of Factors
- 1 is a factor of EVERY number
- Every number is a factor of itself
- A factor is always ≤ the number
- Number of factors is FINITE
Multiples
A multiple is obtained by multiplying the number by any whole number.
Example: Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ...
Properties of Multiples
- Every number is a multiple of itself (×1)
- Multiples of a number are INFINITE
- The smallest multiple of any number is the number itself
- 0 is a multiple of every number (0 × n = 0)
Factor vs Multiple — Know the Difference
| Aspect | Factor | Multiple |
|---|---|---|
| Size | Always ≤ the number | Always ≥ the number |
| Count | Finite | Infinite |
| Example for 6 | 1, 2, 3, 6 | 6, 12, 18, 24, ... |
| Check | 6 ÷ 3 = 2 (exact) | 6 × 3 = 18 |
3. Prime and Composite Numbers
Prime Numbers
A prime number has exactly two factors: 1 and itself.
First 15 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Important: 2 is the ONLY even prime number. All other even numbers are composite (at least factor 2).
Composite Numbers
A composite number has more than two factors.
First few composites: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20
Special Cases
| Number | Factors | Type |
|---|---|---|
| 1 | Only 1 | Neither prime nor composite |
| 2 | 1, 2 | Prime (smallest and only even prime) |
| 0 | Infinite | Neither (undefined for prime/composite) |
Sieve of Eratosthenes (Method to Find Primes)
To find primes up to 100:
- List numbers 1 to 100
- Cross out 1 (not prime)
- Circle 2, cross out all multiples of 2 (4, 6, 8, ...)
- Circle 3, cross out all multiples of 3 (6, 9, 12, ...)
- Circle 5, cross out all multiples of 5
- Circle 7, cross out all multiples of 7
- Remaining circled numbers = primes!
4. Divisibility Tests
These shortcuts tell you if one number divides another WITHOUT performing long division.
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 4,832 → last digit 2 → YES |
| 3 | Sum of digits is divisible by 3 | 4,832: 4+8+3+2=17 → 17 NOT ÷ by 3 → NO |
| 4 | Last TWO digits form a number divisible by 4 | 4,832: last two = 32 → 32 ÷ 4 = 8 → YES |
| 5 | Last digit is 0 or 5 | 4,835 → last digit 5 → YES |
| 6 | Divisible by BOTH 2 AND 3 | 4,836: even (✓2), sum=21 (✓3) → YES |
| 8 | Last THREE digits divisible by 8 | 4,832: 832 ÷ 8 = 104 → YES |
| 9 | Sum of digits divisible by 9 | 4,833: 4+8+3+3=18 → 18 ÷ 9 = 2 → YES |
| 10 | Last digit is 0 | 4,830 → YES |
| 11 | Difference of sum of digits in odd places and even places is 0 or divisible by 11 | 4,839: odd-sum=4+3=7, even-sum=8+9=17, diff=10 → NOT ÷ by 11 → NO |
'Divisibility rules are time-savers in exams. Memorise all 10 rules.'
Worked Example: Test 2,415 for divisibility
| Test | Result |
|---|---|
| 2 | Last digit 5 → NO |
| 3 | 2+4+1+5=12 → 12 ÷ 3 = 4 → YES |
| 4 | Last two = 15 → 15 not ÷ by 4 → NO |
| 5 | Last digit 5 → YES |
| 6 | Must pass both 2 and 3 → fails 2 → NO |
5. Common Factors and Multiples
Common Factors
Factors shared by two or more numbers.
Example: Find common factors of 12 and 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6
- Highest Common Factor (HCF): 6
Common Multiples
Multiples shared by two or more numbers.
Example: Find first three common multiples of 4 and 6
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...
- Common multiples: 12, 24, 36, ...
- Least Common Multiple (LCM): 12
6. Prime Factorisation
Breaking a composite number into its prime factors.
Factor Tree Method
36
/ \
2 18
/ \
2 9
/ \
3 3
36 = 2 × 2 × 3 × 3 = 2² × 3²
Division Method
| Step | Operation |
|---|---|
| 2 | 36 ÷ 2 = 18 |
| 2 | 18 ÷ 2 = 9 |
| 3 | 9 ÷ 3 = 3 |
| 3 | 3 ÷ 3 = 1 |
36 = 2 × 2 × 3 × 3 = 2² × 3²
Practice
| Number | Prime Factorisation |
|---|---|
| 24 | 2³ × 3 |
| 48 | 2⁴ × 3 |
| 56 | 2³ × 7 |
| 72 | 2³ × 3² |
| 100 | 2² × 5² |
| 144 | 2⁴ × 3² |
7. HCF and LCM by Prime Factorisation
Finding HCF
Steps:
- Prime factorise each number
- Identify common prime factors
- Multiply common factors (with smallest power)
Example: HCF of 24 and 36
- 24 = 2³ × 3
- 36 = 2² × 3²
- Common: 2² and 3
- HCF = 2² × 3 = 4 × 3 = 12
Finding LCM
Steps:
- Prime factorise each number
- Take ALL prime factors
- For each prime, take the HIGHEST power
Example: LCM of 24 and 36
- 24 = 2³ × 3
- 36 = 2² × 3²
- Highest powers: 2³ and 3²
- LCM = 8 × 9 = 72
Relation Between HCF and LCM
For any two numbers a and b: a × b = HCF(a,b) × LCM(a,b)
Example: a = 24, b = 36
- LHS: 24 × 36 = 864
- HCF = 12, LCM = 72
- RHS: 12 × 72 = 864 ✓
8. Real-Life Word Problems
HCF Problem (Distribution)
A shopkeeper in Visakhapatnam has 84 mangoes and 56 bananas. He wants to pack them into baskets with the same number of each fruit. What is the greatest number of baskets possible?
Solution: HCF of 84 and 56: 84 = 2² × 3 × 7, 56 = 2³ × 7 Common: 2² × 7 = 4 × 7 = 28 Answer: 28 baskets, each with 84÷28 = 3 mangoes and 56÷28 = 2 bananas.
LCM Problem (Timing)
Traffic lights at two junctions in Vijayawada turn green simultaneously at 8:00 AM. One turns green every 60 seconds, the other every 72 seconds. After how many minutes will they turn green together again?
Solution: LCM of 60 and 72: 60 = 2² × 3 × 5, 72 = 2³ × 3² LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360 seconds = 6 minutes Answer: 8:06 AM
9. Common Mistakes — Fix Them Now
| # | Mistake | Correction |
|---|---|---|
| 1 | Calling 1 a prime number | 1 has only ONE factor. Primes need exactly TWO factors. |
| 2 | Confusing factors and multiples | Factors divide a number; multiples are the number times something |
| 3 | Divisibility rule for 8: only checking last digit | For 8, check last THREE digits |
| 4 | Forgetting that 2 is prime | 2 IS prime — the only even prime |
| 5 | Multiplying wrong in HCF/LCM | HCF uses LOWEST powers; LCM uses HIGHEST powers |
10. Exam Focus
Marks Blueprint
| Question Type | Marks | Topic |
|---|---|---|
| MCQ | 1 | Prime/composite identification |
| Fill in blank | 1 | Divisibility rule |
| Short answer | 2 | Prime factorisation |
| Word problem | 3 | HCF or LCM application |
| Long answer | 4 | Divisibility + factorisation combined |
Quick Self-Test (5 Questions)
Q1: Is 91 a prime number?
<details><summary>Answer</summary>No. 91 = 7 × 13. It is composite.</details>Q2: Check if 1,236 is divisible by 6.
<details><summary>Answer</summary>Even (✓2), sum=1+2+3+6=12 (✓3). Yes, divisible by 6.</details>Q3: Find LCM of 12 and 18.
<details><summary>Answer</summary>12 = 2²×3, 18 = 2×3². LCM = 2²×3² = 36.</details>Q4: Find HCF of 72 and 120.
<details><summary>Answer</summary>72 = 2³×3², 120 = 2³×3×5. HCF = 2³×3 = 24.</details>Q5: Two bells ring at intervals of 15 and 20 minutes. When will they ring together next?
<details><summary>Answer</summary>LCM of 15 and 20: 15=3×5, 20=2²×5. LCM = 2²×3×5 = 60. After 60 minutes.</details>11. Chapter Summary
- Factors: exact divisors of a number; Multiples: number × whole number
- Prime: exactly 2 factors; Composite: 3+ factors; 1: neither
- Divisibility tests for 2, 3, 4, 5, 6, 8, 9, 10, 11
- Prime factorisation: factor tree or division method
- HCF: multiply common prime factors with lowest powers
- LCM: multiply all prime factors with highest powers
- Relation: a × b = HCF × LCM
'Playing with numbers is the first step into the beautiful world of mathematics. These concepts will follow you through every grade.'
