Prime Time — Class 6 Maths (Ganita Prakash)
"Prime numbers are the atoms of mathematics — the building blocks from which all other numbers are made."
1. About This Chapter
Prime Time is one of the most important chapters in Ganita Prakash. It introduces prime numbers — the fundamental building blocks of all numbers — along with factors, multiples, and the game-changing concept of prime factorisation. The chapter uses engaging activities like the "Idli-Vada Game" to make abstract number theory concepts concrete and memorable.
2. Factors
A factor of a number is a number that divides it exactly without leaving a remainder.
For example, the factors of 12 are: 1, 2, 3, 4, 6, 12.
We say:
- 2 is a factor of 12 because 12 ÷ 2 = 6 (exact division)
- 5 is NOT a factor of 12 because 12 ÷ 5 = 2.4 (not exact)
Finding All Factors
To find all factors of 12, check divisibility by 1, 2, 3, ..., 12:
- 12 ÷ 1 = 12 ✓ → 1 and 12 are factors
- 12 ÷ 2 = 6 ✓ → 2 and 6 are factors
- 12 ÷ 3 = 4 ✓ → 3 and 4 are factors
- 12 ÷ 4 = 3 ✓ (already found)
- and so on...
Factors always come in pairs that multiply to give the original number.
3. Multiples
A multiple of a number is the product of that number and any integer.
Multiples of 5: 5, 10, 15, 20, 25, 30, ...
Key points:
- Every number has infinite multiples
- The smallest multiple of a number is the number itself
- A number is a factor of its multiples
4. Common Factors and Common Multiples
Common Factors
Numbers that are factors of two or more given numbers.
Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 Common factors of 12 and 18: 1, 2, 3, 6
Common Multiples
Numbers that are multiples of two or more given numbers.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36... Multiples of 6: 6, 12, 18, 24, 30, 36... Common multiples of 4 and 6: 12, 24, 36...
5. Prime Numbers
A prime number is a number greater than 1 that has exactly two factors: 1 and itself.
The first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
Key facts:
- 2 is the only even prime number — all other primes are odd
- 1 is NOT a prime number (it has only one factor: 1 itself)
- Prime numbers cannot be broken down into smaller factors (except 1 and themselves)
6. Composite Numbers
A composite number has more than two factors.
Examples: 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 12 (factors: 1, 2, 3, 4, 6, 12)
Note: 1 is neither prime nor composite.
7. Prime Factorisation
Every composite number can be written as a product of prime numbers. This is called prime factorisation.
Method: The Factor Tree
Find the prime factorisation of 60:
60
/ \
2 30
/ \
2 15
/ \
3 5
So, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
The Fundamental Idea: Every composite number has a UNIQUE prime factorisation (except for the order of factors).
8. Co-Prime Numbers
Two numbers are co-prime (or relatively prime) if their only common factor is 1.
Examples:
- 8 and 15 — Factors of 8: 1, 2, 4, 8; Factors of 15: 1, 3, 5, 15. Common factor: only 1. ✓ Co-prime!
- 12 and 18 — Common factors: 1, 2, 3, 6. ✗ NOT co-prime.
Note: Co-prime numbers don't have to be prime themselves. 8 and 15 are both composite but co-prime.
9. LCM and GCD (HCF) — Brief Introduction
GCD (Greatest Common Divisor) / HCF (Highest Common Factor)
The largest number that divides two or more given numbers exactly.
For 12 and 18: Common factors: 1, 2, 3, 6. GCD = 6
LCM (Least Common Multiple)
The smallest number that is a multiple of two or more given numbers.
For 4 and 6: Common multiples: 12, 24, 36... LCM = 12
10. The Idli-Vada Game
This is a fun classroom game that teaches multiples:
- Students sit in a circle
- When a number divisible by 3 is called, say "Idli"
- When a number divisible by 5 is called, say "Vada"
- When a number divisible by both 3 AND 5 (i.e., 15, 30, 45...), say "Idli-Vada"
This teaches the concept of common multiples in an engaging, memorable way.
11. Key Concepts Summary
| Concept | Definition | Example |
|---|---|---|
| Factor | Divides the number exactly | Factors of 12: 1, 2, 3, 4, 6, 12 |
| Multiple | Product of number and any integer | Multiples of 5: 5, 10, 15, 20... |
| Prime Number | Exactly two factors: 1 and itself | 2, 3, 5, 7, 11, 13... |
| Composite Number | More than two factors | 4, 6, 8, 9, 10, 12... |
| Co-Prime | Only common factor is 1 | 8 and 15 |
| Prime Factorisation | Expressing a number as product of primes | 60 = 2² × 3 × 5 |
| GCD/HCF | Largest common factor | GCD of 12, 18 = 6 |
| LCM | Smallest common multiple | LCM of 4, 6 = 12 |
12. Important Vocabulary
- Prime: A number >1 with exactly two factors
- Composite: A number >1 with more than two factors
- Factorisation: Breaking a number into its factors
- Co-Prime: Two numbers whose only common factor is 1
- HCF (GCD): Highest Common Factor / Greatest Common Divisor
- LCM: Least Common Multiple
13. Worked Examples
Example 1: Prime or Composite?
Is 37 prime or composite?
Solution: Check divisibility: 37 is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. Its only factors are 1 and 37. So 37 is prime.
Example 2: Prime factorisation of 72
Solution:
72 = 8 × 9
= (2 × 2 × 2) × (3 × 3)
= 2³ × 3²
Example 3: Find GCD and LCM of 12 and 20
Solution:
- Prime factorisation: 12 = 2² × 3, 20 = 2² × 5
- GCD: Take lowest power of common primes → 2² = 4. GCD = 4
- LCM: Take highest power of all primes → 2² × 3 × 5 = 60. LCM = 60
Example 4: Co-prime check
Are 14 and 25 co-prime?
Solution: Factors of 14: 1, 2, 7, 14. Factors of 25: 1, 5, 25. Common factor: only 1. Yes, co-prime.
14. Conclusion
Prime Time introduces the most fundamental concept in number theory: every whole number is either prime or can be expressed uniquely as a product of primes. This idea — the Unique Prime Factorisation Theorem — is one of the pillars of mathematics. Beyond the theory, this chapter builds practical skills: finding factors and multiples, computing GCD and LCM, and recognizing prime and composite numbers. These skills are used extensively in fractions (Chapter 7) and algebra in higher classes.
