Class 7 Maths · Chapter 11
Short answer: NCERT Class 7 Maths (Ganita Prakash) Chapter 11 "Finding Common Ground" HCF aur LCM sikhata hai — factors, multiples, prime factorisation, Highest Common Factor aur Lowest Common Multiple ka concept, unko real-life problems (tile fitting, bell ringing, packet distribution) me apply karna. Ye article poore chapter ke intext questions, exercise solutions, common mistakes aur exam-style questions cover karta hai, saath hi related Ganita Prakash chapters ke liye bhi links deta hai.
Class 7 Ganita Prakash ka Chapter 11 "Finding Common Ground" number theory ka ek bahut practical chapter hai — isme hum seekhte hain ki do ya zyada numbers me kya common hota hai (HCF) aur kya common multiple hota hai (LCM). Ye topic sirf exam ke liye nahi hai — real life me bhi kaam aata hai, jaise tiles fit karna, bells ka ek saath bajna, ya packets equally distribute karna.
Ye article NCERT Class 7 Maths Ganita Prakash chapter wise solutions ki series ka hissa hai, jisme humne har intext aur exercise question ko step-by-step solve kiya hai — CBSE marking scheme ke style me. Agar aap Class 7 Maths ka poora syllabus dekh rahe ho, toh ye chapter Large Numbers Around Us, Arithmetic Expressions, aur Number Play jaise chapters ke baad aata hai, aur isme number sense ki jo neev pehle chapters me banti hai, wahi yahan use hoti hai.
Chapter ka core idea simple hai: Highest Common Factor (HCF) woh sabse bada number hai jo do ya zyada numbers ko poora-poora divide kar sake, aur Lowest Common Multiple (LCM) woh sabse chhota number hai jo un sab numbers ka multiple ho. Chapter me prime factorisation method aur division method dono se HCF-LCM nikalna sikhaya gaya hai, saath hi word problems jisme in concepts ko apply karna padta hai.
Chapter 11 Summary — 5 Minute Revision
Chapter 11 "Finding Common Ground" HCF aur LCM ke fundamentals se shuru hokar unki real-world applications tak jaata hai. Factors aur multiples ka revision, prime factorisation ka tree/ladder method, HCF-LCM nikalne ke do standard tareeke (prime factorisation aur division method), aur inka relationship (HCF × LCM = product of two numbers, jab tak numbers do hi hon) — ye sab chapter ka core hai.
Sabse zyada exam-relevant part word problems hain: kab HCF use karna hai (jaise "sabse bada piece kaatna", "max groups banana") aur kab LCM (jaise "kab dono saath honge", "minimum quantity chahiye"). Students yahan galti karte hain kyunki dono concepts opposite direction me kaam karte hain — HCF divide karta hai, LCM multiply hokar cover karta hai.
In-Text Questions — Solutions
36 aur 60 ke sabhi common factors likho, aur unme se HCF pata karo.
36 ke factors = 1, 2, 3, 4, 6, 9, 12, 18, 36
60 ke factors = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Common factors = 1, 2, 3, 4, 6, 12
Inme sabse bada = 12, isliye HCF(36, 60) = 12.
4 aur 6 ke pehle 5 common multiples likho.
4 ke multiples = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ...
6 ke multiples = 6, 12, 18, 24, 30, 36, 42, 48, ...
Common multiples = 12, 24, 36, 48, 60 (pehle 5)
Inme sabse chhota (LCM) = 12.
Prime factorisation method se 84 ka HCF aur 90 ka LCM nikalne ke liye pehle unka prime factorisation likho.
84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
90 = 2 × 3 × 3 × 5 = 2 × 3² × 5
HCF = common prime factors ka minimum power product = 2 × 3 = 6
LCM = sabhi prime factors ka maximum power product = 2² × 3² × 5 × 7 = 4 × 9 × 5 × 7 = 1260
Do numbers 15 aur 20 ka HCF × LCM nikal kar check karo ki yeh unke product ke barabar hai ya nahi.
15 = 3 × 5, 20 = 2² × 5
HCF(15, 20) = 5, LCM(15, 20) = 60
HCF × LCM = 5 × 60 = 300
15 × 20 = 300
Dono barabar hain, isliye rule confirm hota hai: HCF × LCM = product of two numbers (sirf do numbers ke liye).
Ek dukaandaar ke paas 48 pens aur 60 pencils hain. Unhe barabar-barabar packets me baantna hai bina kuch bache. Max kitne packets ban sakte hain?
Max packets = HCF(48, 60)
48 = 2⁴ × 3, 60 = 2² × 3 × 5
HCF = 2² × 3 = 12
Isliye max 12 packets ban sakte hain — har packet me 4 pens aur 5 pencils.
Exercise Questions — Solutions (Q1–Q6)
Prime factorisation method se 24 aur 36 ka HCF nikalo.
24 = 2 × 2 × 2 × 3 = 2³ × 3
36 = 2 × 2 × 3 × 3 = 2² × 3²
Common prime factors (minimum power) = 2² × 3 = 4 × 3 = 12
HCF(24, 36) = 12
Division method (long division / Euclid method) se 198 aur 429 ka HCF nikalo.
429 = 198 × 2 + 33
198 = 33 × 6 + 0
Jab remainder 0 aata hai, tab jo divisor use hua wahi HCF hai.
Isliye HCF(198, 429) = 33
12, 15 aur 20 ka LCM prime factorisation se nikalo.
12 = 2² × 3
15 = 3 × 5
20 = 2² × 5
LCM = sabhi primes ka max power = 2² × 3 × 5 = 4 × 3 × 5 = 60
Do numbers ka HCF 6 hai aur LCM 72 hai. Agar ek number 24 hai, toh dusra number kya hoga?
Rule: HCF × LCM = Number1 × Number2
6 × 72 = 24 × Number2
432 = 24 × Number2
Number2 = 432 ÷ 24 = 18
Verify: 24 = 2³ × 3, 18 = 2 × 3² → HCF(24, 18) = 2 × 3 = 6 ✓ aur LCM(24, 18) = 2³ × 3² = 72 ✓ — dono diye gaye values se match karte hain.
Ek bagiche me 3 ghantiyan har 20, 30 aur 45 second baad bajti hain. Agar teeno ek saath 12:00 baje bajein, toh agli baar teeno saath kab bajengi?
Teeno ka saath bajna LCM se milta hai.
20 = 2² × 5, 30 = 2 × 3 × 5, 45 = 3² × 5
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 seconds
180 seconds = 3 minute
Isliye teeno ghantiyan agli baar 12:03 baje saath bajengi.
Do ropes ki length 72 m aur 120 m hai. Unhe barabar lambai ke pieces me kaatna hai bina waste kiye, taaki har piece ki length maximum ho. Har piece ki length kya hogi, aur kitne pieces banenge?
Max piece length = HCF(72, 120)
72 = 2³ × 3², 120 = 2³ × 3 × 5
HCF = 2³ × 3 = 24
Har piece = 24 m
Total pieces = (72 ÷ 24) + (120 ÷ 24) = 3 + 5 = 8 pieces
Important Equations — Ek Nazar Me
| Concept | Formula / Rule |
|---|---|
| HCF (Highest Common Factor) | Sabse bada number jo diye gaye sab numbers ko poora divide kare |
| LCM (Lowest Common Multiple) | Sabse chhota number jo diye gaye sab numbers ka multiple ho |
| Prime factorisation se HCF | Common prime factors ka minimum power ka product |
| Prime factorisation se LCM | Sabhi prime factors ka maximum power ka product |
| Do numbers ka relation | HCF × LCM = Number1 × Number2 (sirf do numbers ke liye) |
| Kab HCF use karo | Max piece/group banane wale, barabar baantne wale problems me |
| Kab LCM use karo | Saath hona, repeat hona, minimum common quantity wale problems me |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- HCF aur LCM ke use-case ko confuse karna — 'saath milna/repeat hona' wale problems me HCF laga dena jabki wahan LCM chahiye, aur 'max piece/group' wale problems me LCM laga dena.
- Prime factorisation karte waqt kisi prime factor ko miss kar dena, especially bade numbers me — isse pura HCF/LCM galat aa jaata hai.
- HCF nikalte waqt common prime factors ka maximum power le lena (jabki minimum lena chahiye) — ye sabse common calculation error hai.
- LCM nikalte waqt kisi prime factor ka sirf ek hi power le lena, jab wo do numbers me alag-alag power se present ho — hamesha maximum power lena chahiye.
- HCF × LCM = product wala formula teen ya zyada numbers pe apply kar dena — ye formula sirf do numbers ke liye valid hai.
- Division method (Euclid method) me remainder 0 aane tak process continue na karna, ya beech me hi galat divisor ko HCF maan lena.
Board-Style Important Questions
- HCF aur LCM prime factorisation method se nikalne wale do-number ya teen-number problems — school tests aur term exams me common
- Word problems jisme 'ek saath bajna/milna/repeat hona' scenario ho (bells, traffic lights, lapping) — LCM based
- Word problems jisme 'max/largest piece, group, ya packet' banane ho bina waste kiye — HCF based
- HCF × LCM = product of numbers wala relation use karke missing number nikalna
- Division (Euclid) method se HCF nikalne wale step-by-step questions
- Real-life application questions jisme student ko khud decide karna ho ki HCF chahiye ya LCM
Aksar Poochhe Jaane Wale Sawaal
NCERT Class 7 Maths Chapter 11 'Finding Common Ground' me kya sikhaya jaata hai?
Is chapter me HCF (Highest Common Factor) aur LCM (Lowest Common Multiple) ka concept sikhaya jaata hai — inko prime factorisation aur division method se nikalna, aur real-life word problems me apply karna.
HCF aur LCM me kya farak hai?
HCF sabse bada common factor hota hai jo numbers ko divide karta hai, jabki LCM sabse chhota common multiple hota hai jo un numbers se divide ho jaata hai. HCF divide karne wale problems me kaam aata hai, LCM 'saath hona/repeat' wale problems me.
HCF × LCM = product of two numbers wala formula kab use hota hai?
Ye formula sirf tab valid hai jab sirf DO numbers ho. Teen ya zyada numbers ke liye ye formula seedha apply nahi hota — HCF/LCM alag se prime factorisation se nikalni padti hai.
Kaise pata karein kisi word problem me HCF chahiye ya LCM?
Agar problem 'barabar baantna', 'max piece kaatna', 'sabse bada group banana' jaisa hai — HCF use karo. Agar problem 'kab saath hoga', 'kab repeat hoga', 'minimum quantity chahiye jo sab me fit ho' jaisa hai — LCM use karo.
Prime factorisation method HCF/LCM nikalne ka best tareeka kyun mana jaata hai?
Kyunki isme har number ko uske prime factors me tod diya jaata hai, jisse common factors (HCF ke liye) aur sabhi zaroori factors (LCM ke liye) clearly dikh jaate hain — chhoti galti ki gunjaish kam hoti hai bade division method ke comparison me.
Class 7 Ganita Prakash ke doosre chapters is chapter se kaise connected hain?
Chapter 11 ka number sense Chapter 1 'Large Numbers Around Us' aur Chapter 3 'Number Play' se aage badhta hai. HCF/LCM ka concept aage Chapter 8 'Working with Fractions' me bhi kaam aata hai, jahan fractions ko simplify karne ya common denominator nikalne ke liye HCF-LCM use hote hain.
Class 7 Maths — Saare Chapters
NCERT Kaksha ek swatantra shaikshik platform hai aur NCERT ya CBSE se aadhikarik roop se sambaddh (officially affiliated) nahi hai. Kisi galti ki jaankari dene ke liye contact kijiye.