Complete Summary, Explanations, and Solutions for Prime Time – Ganita Prakash Class VI, Chapter 5 – Factors, Multiples, Prime Numbers, Co-primes, Questions, Answers
Detailed summary and explanation of Chapter 5 'Prime Time' from the Ganita Prakash Mathematics textbook for Class VI, covering common multiples and common factors, prime and composite numbers, Sieve of Eratosthenes, co-prime numbers, prime factorization, divisibility tests for 2, 3, 4, 5, 8, and 10, twin primes, perfect numbers—along with all NCERT questions, answers, and step-by-step solutions.
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Prime Time
NCERT Class 6 Mathematics Chapter 5 | Complete Guide | Prime Numbers 2025
Chapter at a Glance – Prime Time
This chapter explores prime numbers as building blocks, common multiples/factors through games, and co-primes. Includes visualizations like Sieve and thread art.
Main Topics Covered
- Common multiples & factors: Idli-Vada game, LCM/HCF concepts.
- Factors & divisors: Jump Jackpot game.
- Prime numbers: Definition, Sieve of Eratosthenes, identification.
- Composite numbers & perfect numbers.
- Co-prime numbers: Safe pairs, relation to LCM=product.
- Visualizations: Rectangular arrangements, thread art.
Key Takeaways for Exams
Multiples
Numbers like 3x: 3,6,9,... Common: LCM.
Factors
Divisors of number, e.g., 24: 1,2,3,4,6,8,12,24. Common: HCF.
Primes
Only 1 & itself factors: 2,3,5,7,11,...
Composites
More than two factors: 4,6,8,9,...
Co-primes
HCF=1, e.g., 4&9. LCM=product.
Perfect Numbers
Sum factors=2x number, e.g., 6,28.
Sieve
Method to find primes up to n.
Twin Primes
Difference 2: 3&5,5&7,...
Expanded: Learn through games for better retention. Practice listing primes 1-100.
Key Rules & Properties – Prime Time
Essential rules for multiples, factors, primes, co-primes.
Multiples & Factors Rules
Definitions and relations.
| Concept | Rule/Property | Example |
|---|---|---|
| Multiple | n × k for integer k | Multiples of 3: 3,6,9,12 |
| Common Multiple | Least is LCM | LCM(3,5)=15 |
| Factor | Divides exactly | Factors 24: 1,2,3,4,6,8,12,24 |
| Common Factor | Greatest is HCF/GCD | HCF(14,36)=2 |
| Perfect Number | Sum factors = 2×number | 28: 1+2+4+7+14+28=56 |
Prime & Composite Properties
| Property | Description | Example |
|---|---|---|
| Prime | Exactly two factors: 1 & itself | 2,3,5,7 |
| Composite | More than two factors | 4=2×2,6=2×3 |
| 1 is Neither | Only one factor | 1 |
| Even Prime | Only 2 | All others odd |
| Twin Primes | Difference 2 | 3&5,11&13 |
Co-prime Properties
- Definition: HCF=1.
- LCM Relation: LCM(a,b)=a×b if co-prime.
- Examples: 4&9,15&37.
- Thread Art: Complete loop if co-prime pegs & gap.
Expanded: LCM×HCF = a×b always. Use for problem-solving.
Concept Cards – Quick Explanations
Common Multiples
Shared multiples, LCM smallest non-zero.
Common Factors
Shared divisors, HCF largest.
Prime Numbers
Two factors only, building blocks.
Composite Numbers
Multiple factors, product of primes.
Perfect Numbers
Sum proper factors = number.
Co-primes
No common factors except 1.
Sieve of Eratosthenes
Method to list primes.
Twin Primes
Primes differing by 2.
Idli-Vada Game
Teaches multiples/LCM.
Jump Jackpot
Teaches factors/HCF.
Co-prime Art
Thread patterns with pegs.
Rectangular Arrangements
Visualize factors.
Expanded: Use games for intuitive understanding.
Examples + Solutions
Example 1: Common Multiples of 3 & 5
Solution: 15,30,45,... LCM=15.
Example 2: Factors of 24
Solution: 1,2,3,4,6,8,12,24.
Example 3: HCF of 14 & 36
Solution: 2 (common:1,2).
Example 4: Is 28 Perfect?
Solution: Yes, 1+2+4+7+14=28, sum factors=56=2×28.
Example 5: Co-prime Check 4 & 9
Solution: Yes, HCF=1. LCM=36=4×9.
Example 6: Primes 21-30
Solution: 23,29. Composites:21,22,24,25,26,27,28,30.
Example 7: Twin Primes <20
Solution: (3,5),(5,7),(11,13),(17,19).
Additional Example: LCM of 4,6,8
Solution: 24.
Figure it Out Solutions (All Solved)
5.1 Common Multiples & Factors
1. 'Idli-vada' 10th time?
Ans. 150 (LCM 3&5=15, 15×10=150).
2a. 'Idli' times to 90 (incl idli-vada).
Ans. 30 (90/3=30).
2b. 'Vada' times to 90.
Ans. 18 (90/5=18).
2c. 'Idli-vada' times to 90.
Ans. 6 (90/15=6).
3. To 900?
Ans. Idli:300, Vada:180, Idli-vada:60.
4. Figure related to game?
Ans. Yes, Venn diagram for common multiples. To 60: Double circles.
Other number for game: 2,3,5,8,10?
Ans. 4 (yesterday's numbers).
Jump sizes for 24?
Ans. 1,2,3,4,6,8,12,24.
Jump for 14&36?
Ans. 1,2 (HCF=2).
Jump for 15&30?
Ans. 1,3,5,15.
Table: Shaded common?
Ans. Multiples of 3&5? (Assume shaded multiples 3, circled 5, both LCM).
1. Multiples 40 between 310-410.
Ans. 320,360,400.
2a. Number <40, factor 7, digits sum 8.
Ans. 35 (3+5=8).
2b. Number <100, factors 3&5, digits differ by 1.
Ans. 45 (4&5 differ 1),75 (7&5 differ 2? Wait, 45).
3. Perfect between 1-10.
Ans. 6 (1+2+3=6).
4a. Common factors 20&28.
Ans. 1,2,4.
4b. 35&50.
Ans. 1,5.
4c. 4,8,12.
Ans. 1,2,4.
4d. 5,15,25.
Ans. 1,5.
5. Multiples 25 not 50.
Ans. 25,75,125.
6. Idli-vada first after 50, <10 numbers.
Ans. 7&8 (LCM=56>50).
7. Jump 28&70.
Ans. 1,2,7,14 (HCF=14).
8. Erased common multiples diagram.
Ans. Assume multiples a=24, b=48, common 24,48,72,... Fill others.
9. Smallest multiple 1-10 except 7.
Ans. LCM(1-6,8-10)=2520/7? Wait, 360 (check).
10. Smallest multiple 1-10.
Ans. 2520.
5.2 Prime Numbers
Guna's arrangements for 12 figs.
Ans. 1x12,2x6,3x4,4x3,6x2,12x1.
Anshu's for 7.
Ans. 1x7,7x1 (prime).
Primes 21-30.
Ans. 23,29 (2 primes).
Composites 21-30.
Ans. 21,22,24,25,26,27,28,30 (8).
1. Even prime?
Ans. Only 2.
2. Smallest/largest difference successive primes <100.
Ans. Smallest 2 (twins), largest 8 (113-107? <100:89-97? 97-89=8).
3. Primes per row in table.
Ans. No equal. Decades: 1-10:4, 11-20:4, etc. Least 90s:2, most 1-10:4.
4. Primes:23,51,37,26?
Ans. 23,37 prime;51=3x17,26=2x13 composite.
5. Prime pairs <20 sum multiple 5.
Ans. (2,3)=5,(2,13)=15,(2,18 no),(3,7)=10 etc.
6. Pairs like 13&31 <100.
Ans. 17&71,37&73,79&97 etc.
7. Seven consecutive composites <100.
Ans. 90-96:90,91,92,93,94,95,96.
8. Twin primes <100.
Ans. (3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61),(71,73).
9. Statements true/false.
Ans. a. False (units 2,3,5,7). b. False (product composite). c. False (have 1&itself). d. False (2 prime). e. True (next after prime >3 even, composite).
10. Product exactly three distinct primes.
Ans. 105=3×5×7.
11. Three-digit primes from 2,4,5 each once.
Ans. 245 no,425 no,524 no,542 no,425 no. None prime.
12. 2p+1 prime for prime p.
Ans. p=3:7, p=5:11, p=11:23, p=17:35 no, p=19:39 no. Examples:3,5,11.
5.3 Co-prime Numbers
Safe pairs:15&39,4&15,18&29,20&55.
Ans. a. No (HCF=3), b. Yes (1), c. Yes (1), d. No (5).
Co-prime pairs.
Ans. a. Yes (1), b. Yes, c. No (5), d. No (17? Wait 17&69=17×4+1, HCF=1 yes), e. No (9).
1-2. Idli-vada observations.
Ans. 1. Co-prime: LCM=product e.g. 3&4=12. 2. Not: LCM
Co-prime art observations.
Ans. Complete if pegs & gap co-prime (e.g. 13&3).
Extra Practice Questions (Exam-Ready) – Chapter 5
35+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags
1-Mark Questions (Very Short Answer)
1. LCM of 2&5.
2. HCF of 12&18.
3. Is 9 prime?
4. Factors of 30.
5. Co-prime:8&15?
6. Even prime.
7. Twin primes example.
8. Perfect number <10.
2-Mark Questions (Short Answer)
9. Multiples of 4 between 20-40.
10. Common factors 24&36.
11. Primes 40-60.
12. LCM 6&8.
13. Co-prime pairs <20.
14. Sum factors 6.
3-Mark Questions (Reasoning)
15. Why 1 neither prime/composite?
16. Explain Sieve for primes <20.
17. Why co-prime LCM=product?
18. Rectangular arrangements 18.
19. Consecutive composites 8.
20. 2p+1 prime examples.
4–5 Mark Questions (Application)
21. Smallest multiple 1-12 except 11.
22. Idli-vada to 120: counts.
23. Factors 100.
24. Co-prime art for 10 pegs gap 3.
25. Product three distinct primes <200.
Challenge Questions (6+ Marks)
26. Prove no primes end with 4.
27. Find 10 twin primes <200.
28. Sieve diagram <50.
29. LCM HCF relation proof small.
30. Perfect number next after 28.
31. Co-prime examples 5 pairs.
32. Jump sizes for three numbers.
33. Why infinite primes?
34. Goldbach conjecture small.
35. Mersenne primes.
Common Mistakes & How to Avoid
Mistake 1: 1 as Prime
Classifying 1 as prime.
Avoid: Remember 1 has one factor.
Mistake 2: Even Primes >2
Thinking even numbers prime.
Avoid: All even >2 divisible by 2.
Mistake 3: LCM/HCF Mixup
Confusing least/greatest.
Avoid: LCM multiple, HCF factor.
Mistake 4: Missing Factors
Forgetting 1 & number itself.
Avoid: Always include.
Mistake 5: Co-prime with Common 1 Only
Ignoring check.
Avoid: List factors.
Mistake 6: Sieve Errors
Crossing wrong multiples.
Avoid: Start from 2, cross multiples.
Mistake 7: Perfect Sum Include Number?
Including number in proper factors.
Avoid: Proper exclude itself, but definition sum all=2x.
Expanded: Practice listing for small numbers.
History & Fun Facts
Ancient Origins
Sieve by Eratosthenes ~200 BC Greece.
Primes studied by Euclid (infinite primes proof).
Perfect numbers linked to Mersenne primes.
Real-Life Applications
- Cryptography: Primes in RSA encryption.
- Computing: Factorization hard for security.
- Nature: Cicadas cycles prime years.
- Art: Co-prime thread designs.
Fun Facts
- Largest known prime: 2^82,589,933 -1 (millions digits).
- 2 is only even prime.
- Twin primes conjecture: Infinite pairs.
- Goldbach: Every even >2 sum two primes (unproven).
- Perfect numbers even, none odd known.
- Primes in music rhythms, poetry.
Did You Know?
Virahanka numbers relate? No, but primes fundamental.
Expanded: Mersenne primes for largest known.
Quick Revision One-Pager
Key Concepts
| Term | Quick Note |
|---|---|
| LCM | Least common multiple |
| HCF | Highest common factor |
| Prime | 2 factors |
| Composite | >2 factors |
| Co-prime | HCF=1 |
| Perfect | Sum factors=2x |
Quick Rules
- ✓ Primes: Sieve cross multiples.
- ✓ Factors: Divide exactly.
- ✓ Multiples: Multiply integers.
- ✓ Co-prime: Product=LCM.
- ✓ Twin: Diff 2 primes.
- ✓ Arrangements: Factor pairs.
Mind Map
Central: Numbers
- Multiples/Factors: Common, games
- Primes:
- Definition, Sieve
- Twin, perfect
- Co-primes: Safe, art
Exam Tips
Before Solving
List factors/multiples
During Solving
Use HCF/LCM formulas
After Solving
Verify division
Time-Savers
Prime factorization
Expanded: Memorize primes <100.
Interactive Quiz – 20 Questions
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