Class 6 Maths · Chapter 5
Short answer: Class 6 Maths Ganita Prakash Chapter 5 "Prime Time" prime numbers, factors, multiples, prime factorisation, HCF-LCM, aur Goldbach's Conjecture cover karta hai — sabse important number-sense chapter jo aage ke fractions aur algebra ki neev banata hai.
Class 6 Maths Ganita Prakash ke Chapter 5 "Prime Time" me hum numbers ki duniya ke sabse fundamental building blocks — prime numbers — ke baare me deep dive karte hain. Jaise atoms se pura matter banta hai, waise hi prime numbers se har composite number banta hai. Ye chapter poore Class 6 se aage tak — fractions, ratios, algebra sab me — kaam aane wale number-sense ki neev hai.
Is chapter me factors aur multiples, prime aur composite numbers ki pehchaan, prime factorisation (factor tree method se), divisibility rules, co-prime numbers, aur HCF-LCM nikalna cover hota hai. Sabse interesting part hai Goldbach's Conjecture — ek 280 saal purana unsolved mathematical mystery jo students ko batata hai ki maths me aaj bhi naye discoveries ho sakte hain.
Chapter 5 Summary — 5 Minute Revision
Chapter 5 "Prime Time" Class 6 Ganita Prakash ka ek core number-sense chapter hai jisme students factors, multiples, prime/composite numbers, prime factorisation, divisibility rules, co-prime numbers, aur HCF-LCM seekhte hain. Sabse unique addition hai Goldbach's Conjecture — jo NCERT ne pehli baar Class 6 level pe introduce kiya hai taaki students ko pata chale ki maths me aaj bhi unsolved problems exist karte hain.
Prime factorisation ka concept sirf is chapter tak limited nahi hai — Class 6 ke aage Fractions (simplification), Ratio-Proportion, aur higher classes ke Algebra tak isi neev pe building hoti hai. Isliye is chapter ko sirf ratta maar ke nahi, concept clear karke padhna chahiye — especially factor tree method aur HCF/LCM ka prime factorisation approach jo NCERT is naye rationalised syllabus me strongly emphasize karta hai.
In-Text Questions — Solutions
Figure it Out (Page-based): List all factors of 36 and identify which are prime.
36 ke factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Inme se prime factors sirf: 2 aur 3 hain (kyunki inke sirf 2 factors hain — 1 aur khud).
Figure it Out: Check whether 2, 3, 5, 7 are all factors of 210.
210 ÷ 2 = 105 ✓, 210 ÷ 3 = 70 ✓, 210 ÷ 5 = 42 ✓, 210 ÷ 7 = 30 ✓
Haan, 2, 3, 5, aur 7 chaaron 210 ke factors hain. Isliye 210 = 2 × 3 × 5 × 7 bhi likha ja sakta hai.
Think and Discuss: Why is 1 neither prime nor composite?
Prime number ki definition kehti hai ki uske exactly 2 different factors hone chahiye (1 aur khud number). Lekin 1 ka sirf ek hi factor hai — khud 1 (kyunki 1 aur khud '1' same hain).
Composite number ki definition kehti hai 2 se zyada factors chahiye — 1 ke paas woh bhi nahi hain.
Isliye 1 na prime hai na composite — ye ek special (unique) number hai.
Exercise Questions — Solutions (Q1–Q6)
Q1. Write all the factors of 68.
68 ke saare factors nikalne ke liye hum 1 se 68 tak un numbers ko check karte hain jo 68 ko poora divide karte hain.
68 = 1 × 68 = 2 × 34 = 4 × 17
Factors of 68 = 1, 2, 4, 17, 34, 68
Q2. Find the prime factorisation of 240 using the factor tree method.
240 ko baar-baar chhote prime numbers se divide karte hain jab tak sirf primes na bach jayein.
240 = 2 × 120 = 2 × 2 × 60 = 2 × 2 × 2 × 30 = 2 × 2 × 2 × 2 × 15 = 2 × 2 × 2 × 2 × 3 × 5
240 = 24 × 3 × 5
Q3. Is 337 a prime number? Check using the trial division method.
Kisi number N ko prime check karne ke liye hum sirf un primes se divide karke dekhte hain jinka square N se kam ya barabar ho. Yahan √337 ≈ 18.4, isliye 2, 3, 5, 7, 11, 13, 17 se check karenge.
337 ÷ 2 = not divisible (odd), 337 ÷ 3 → 3+3+7=13, not divisible
337 ÷ 7 = 48.14..., 337 ÷ 11 = 30.6..., 337 ÷ 13 = 25.9..., 337 ÷ 17 = 19.8...
Koi bhi prime 337 ko poora divide nahi karta.
337 ek prime number hai.
Q4. Express 60 as the sum of two primes in two different ways (Goldbach's Conjecture check).
60 ek even number hai (>2), isliye Goldbach's Conjecture ke according ise do primes ke sum se likha ja sakta hai.
60 = 7 + 53
60 = 13 + 47
Do valid ways: 60 = 7 + 53 = 13 + 47 (aur bhi combinations possible hain jaise 29 + 31)
Q5. Find the HCF and LCM of 36 and 90 using prime factorisation.
Step 1: Prime factorisation nikalo
36 = 2² × 3²
90 = 2 × 3² × 5
Step 2: HCF = common primes ka minimum power
HCF = 2¹ × 3² = 2 × 9 = 18
Step 3: LCM = saare primes ka maximum power
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180
HCF = 18, LCM = 180 (check: HCF × LCM = 18 × 180 = 3240 = 36 × 90 ✓)
Q6. A gardener has 48 rose plants and 60 marigold plants. He wants to arrange them in rows such that each row has the same number of plants and only one type of plant. Find the maximum number of plants in each row.
Ye HCF ka application hai kyunki hume equal aur maximum grouping chahiye.
48 = 2⁴ × 3
60 = 2² × 3 × 5
HCF = 2² × 3 = 12
Har row me maximum 12 plants ho sakte hain. Rose ki 4 rows (48÷12) aur marigold ki 5 rows (60÷12) banengi.
Important Equations — Ek Nazar Me
| Concept | Formula / Rule | Example |
|---|---|---|
| Factor | a, b ka factor hai agar b ÷ a se remainder 0 aaye | 4, 20 ka factor hai kyunki 20 ÷ 4 = 5 |
| Prime Number | Sirf 2 factors — 1 aur khud number | 2, 3, 5, 7, 11, 13... |
| Composite Number | 2 se zyada factors | 4, 6, 8, 9, 10... |
| Co-prime Numbers | Do numbers jinka HCF = 1 | 8 aur 15 co-prime hain |
| Prime Factorisation | Number ko sirf prime factors ke product ke roop me likhna | 36 = 2² × 3² |
| HCF (Prime Factorisation Method) | Common primes ka minimum power ka product | HCF(36,90) = 2×3² = 18 |
| LCM (Prime Factorisation Method) | Saare primes (common + non-common) ka maximum power ka product | LCM(36,90) = 2²×3²×5 = 180 |
| HCF × LCM Relation | HCF(a,b) × LCM(a,b) = a × b | 18 × 180 = 36 × 90 = 3240 |
| Goldbach's Conjecture | Har even number >2 = do primes ka sum | 60 = 7+53 = 13+47 |
| Divisibility by 2 | Last digit even (0,2,4,6,8) | 68 → last digit 8 → divisible |
| Divisibility by 3 | Digits ka sum 3 se divisible ho | 337 → 3+3+7=13 → not divisible |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- 1 ko prime number maan lena — jabki 1 ke paas sirf ek factor hai, isliye woh na prime hai na composite.
- Prime factorisation karte waqt beech me hi rukna — jaise 24 = 2 × 2 × 6 likh dena, jabki 6 khud composite hai aur ise aage 2×3 tak todna zaroori hai.
- HCF aur LCM me confuse ho jaana — HCF ke liye minimum power lena chahiye lekin students maximum power le lete hain (ya vice versa).
- Divisibility rule of 3 galat apply karna — poore number ko 3 se divide karne ki koshish karna instead of digits ka sum check karna.
- Co-prime numbers ko prime numbers samajh lena — co-prime ka matlab hai HCF=1, jabki dono numbers khud composite bhi ho sakte hain jaise 8 aur 15.
- Goldbach's Conjecture ko odd numbers pe bhi apply karne ki koshish karna — ye rule sirf even numbers (greater than 2) ke liye hi hai.
Board-Style Important Questions
- Find the HCF and LCM of two numbers using the prime factorisation method and verify the relationship HCF × LCM = product of the numbers.
- Express a given even number as the sum of two prime numbers (Goldbach's Conjecture based question).
- Find the prime factorisation of a 3-digit composite number using the factor tree method.
- Two bells ring at intervals of 12 minutes and 18 minutes. If both ring together at 9:00 AM, find when they will ring together again (LCM-based word problem).
- Determine whether a given large number (e.g., 187 or 221) is prime or composite using the trial division method.
Aksar Poochhe Jaane Wale Sawaal
Class 6 Maths chapter 5 Prime Time me kya-kya topics cover hote hain?
Ganita Prakash ke Chapter 5 'Prime Time' me factors aur multiples, prime aur composite numbers, prime factorisation (factor tree method), divisibility rules, co-prime numbers, HCF aur LCM (prime factorisation method se), aur ek naya interesting topic — Goldbach's Conjecture — cover hota hai. Ye ek unsolved problem hai jo bataata hai ki har even number 2 se bada 2 primes ka sum hota hai, aur NCERT ne isse students ko 'unsolved problems in mathematics' se introduce karwane ke liye add kiya hai.
Goldbach's Conjecture kya hai aur ye important kyun hai?
Goldbach's Conjecture kehta hai ki 2 se bada har even number do prime numbers ka sum hota hai — jaise 10 = 3+7 ya 5+5. Ye 1742 se aaj tak proved nahi hua hai, isliye ye mathematics ka ek famous unsolved problem hai. NCERT isko is chapter me isliye include karta hai taaki students samjhein ki maths sirf 'solved formulas' nahi hai — aaj bhi bahut si cheezein research ke under hain, aur students khud bhi patterns explore kar sakte hain.
Kya Class 6 ki NCERT Maths book ka naam badal gaya hai?
Haan. Purani 14-chapter 'Mathematics' textbook ki jagah ab NEP-aligned rationalised syllabus me 'Ganita Prakash' naam ki nayi book aa gayi hai jisme sirf 10 chapters hain. Is naye syllabus me 'The Other Side of Zero' (negative numbers/integers) jaisa naya chapter add hua hai aur data handling ka scope bhi expand kiya gaya hai.
NCERT Class 6 Maths Ganita Prakash ki PDF kahan se download karein?
Official aur sabse reliable source NCERT ki apni website ncert.nic.in hai, jahan se 2026-27 session ki current rationalised Ganita Prakash textbook free PDF format me download ki ja sakti hai chapter-wise. Kisi bhi third-party site se download karne se pehle ye zaroor confirm karein ki content current rationalised syllabus se match kar raha hai, kyunki purani 'Mathematics' book ke PDFs abhi bhi internet par circulate ho rahe hain.
Prime Time chapter me HCF aur LCM nikalne ka sabse aasan tarika kya hai?
Is chapter me prime factorisation method sabse recommended tarika hai. Pehle dono numbers ko unke prime factors me todo, phir HCF ke liye common primes ka minimum power lo aur LCM ke liye saare primes (common + non-common) ka maximum power lo. Ye method long division method se zyada systematic hai aur bade numbers ke liye bhi easily kaam karta hai — jaisa Q5 aur Q6 me step-by-step dikhaya gaya hai.
Class 6 Maths ke other chapters — jaise Fractions, Symmetry, Lines and Angles — Prime Time se kaise related hain?
Ganita Prakash ki poori 10-chapter sequence ek doosre pe build hoti hai. Chapter 1 'Patterns in Mathematics' number patterns ka intro deta hai jo Chapter 5 'Prime Time' me prime numbers ke patterns tak lekar jaata hai. Baad ke chapters jaise Fractions, Perimeter and Area, Symmetry, aur Lines and Angles alag-alag geometry/arithmetic skills build karte hain, lekin factorisation aur number sense ki jo neev Prime Time me banti hai woh puri book me — especially fraction simplification aur ratio-based problems me — kaam aati rehti hai.
Class 6 Maths — Saare Chapters
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