Class 8 Maths · Chapter 3
Short answer: NCERT Class 8 Ganita Prakash Chapter 3 "A Story of Numbers" mein divisibility rules, prime factorisation, HCF-LCM, Euclid's division lemma, aur Baudhayana-Pythagoras theorem cover hota hai — full step-by-step exercise solutions, formulas table aur common mistakes ke saath.
Class 8 Maths ka Chapter 3, "A Story of Numbers", poori number theory ki neev hai jo aage Class 9-10 tak carry hoti hai. Ye chapter naye Ganita Prakash series (NCERT ka 2026-27 session ke liye rationalised do-part format) ka hissa hai, jismein numbers ki historical journey — divisibility, factors, prime numbers, HCF-LCM se lekar ancient Indian mathematics ke contribution (Baudhayana Sulba Sutra) tak — cover hoti hai.
Ye chapter sirf calculation practice nahi hai — ye samjhata hai ki numbers ka "story" kaise evolve hui, aur kaise Bharat ke ancient mathematicians ne modern geometry ke concepts ko centuries pehle hi derive kar liya tha. Neeche di gayi solutions guide step-by-step CBSE marking-scheme style mein hai, taaki concept clarity aur exam-ready presentation dono mile.
Chapter 3 Summary — 5 Minute Revision
"A Story of Numbers" Class 8 Maths ka wo chapter hai jo numbers ki duniya ko sirf calculation ke roop mein nahi, balki ek historical aur logical journey ke roop mein present karta hai. Divisibility rules se lekar Euclid's Division Lemma tak, aur Baudhayana ke ancient contribution se lekar Pythagorean triplets tak — har concept ek dusre se connect hota hai number theory ki neev banane ke liye.
Is chapter ki strong understanding sirf Class 8 ke exams tak limited nahi hai — HCF-LCM aur prime factorisation jaise concepts Class 9-10 ki Algebra aur Number Systems mein directly reuse hote hain. Isliye har exercise question ko step-by-step samajhna aur common mistakes se bachna important hai, jaisa upar diye gaye solutions aur mistakes list mein cover kiya gaya hai.
In-Text Questions — Solutions
Figure it out (Page shuruaati) — Kya 3-digit numbers mein sabse zyada koi ek digit repeat hoti hai?
Ye ek exploratory activity hai jahan students ko different 3-digit numbers likh kar unke digit patterns observe karne hote hain. NCERT ka intent yahan discovery-based learning hai — koi fixed 'correct' numeric answer nahi, balki pattern-recognition ka practice hai.
Kya ek number sirf ek hi tarike se prime factors mein likha ja sakta hai?
Haan — ye Fundamental Theorem of Arithmetic kehlata hai: har composite number ko prime factors ke product ke roop mein sirf ek hi unique tarike se likha ja sakta hai (order ignore karke).
Example: 60 = 2² × 3 × 5 — is order ke alawa koi aur combination possible nahi hai.
Do consecutive numbers ka HCF hamesha kya hota hai?
Do consecutive numbers (jaise 15 aur 16) hamesha co-prime hote hain, matlab unka HCF hamesha 1 hota hai — kyunki unmein koi common prime factor possible nahi hota.
Exercise Questions — Solutions (Q1–Q6)
Q1. Check whether the following numbers are divisible by 11: (a) 1331 (b) 61809 (c) 90728
Divisibility rule of 11: Sum of digits at odd places − sum of digits at even places (from right) honi chahiye 0 ya 11 ka multiple.
(a) 1331 → (1+3) − (3+1) = 4 − 4 = 0 → divisible by 11 ✓
(b) 61809 → (9+8+6) − (0+1) = 23 − 1 = 22 → 22 = 11 × 2 → divisible by 11 ✓
(c) 90728 → (8+7+0) − (2+9) = 15 − 11 = 4 → not divisible by 11 ✗
Q2. Find the HCF of 96 and 404 by prime factorisation method, aur unka LCM bhi nikaalo.
Step 1 — Prime factorisation:
96 = 2⁵ × 3
404 = 2² × 101
Step 2 — HCF = common prime factors ka lowest power = 2² = 4
Step 3 — LCM formula se: LCM = (96 × 404) ÷ HCF
LCM = 38784 ÷ 4 = 9696
Q3. Euclid ki division lemma use karke HCF(867, 255) nikaalo.
Euclid's Division Algorithm: a = bq + r, jab tak r = 0 na ho jaaye.
867 = 255 × 3 + 102
255 = 102 × 2 + 51
102 = 51 × 2 + 0
Jab remainder 0 aata hai, us step ka divisor hi HCF hota hai.
HCF(867, 255) = 51
Q4. Ek Pythagorean triplet find karo jismein ek number 18 ho.
Standard generator formula: agar 2m diya ho, to triplet hota hai (2m, m²−1, m²+1).
2m = 18 → m = 9
m² − 1 = 81 − 1 = 80
m² + 1 = 81 + 1 = 82
Check: 18² + 80² = 324 + 6400 = 6724 = 82² ✓
Triplet: (18, 80, 82)
Q5. Baudhayana Sulba Sutra ke context mein bataiye ki right-angled triangle ka rule kya tha, aur ise aaj kaunse theorem se jaana jaata hai?
Baudhayana Sulba Sutra (~800 BCE) mein diya gaya tha ki ek rassi (cord) jo right-angled triangle ke hypotenuse ke barabar length ki ho, uska square area us area ke barabar hota hai jo do sides — base aur height — dwara bane squares ka sum hota hai.
Hypotenuse² = Base² + Height²
Yehi rule aaj Baudhayana–Pythagoras theorem ke naam se jaana jaata hai, kyunki iska likhit pramaan Pythagoras se pehle Bharat mein already maujood tha.
Q6. 2, 3, aur 5 dono se divisible sabse chhota 4-digit number find karo.
2, 3, 5 se divisible hone ke liye number 30 (= 2×3×5) ka multiple hona chahiye — kyunki ye teeno co-prime hain, isliye LCM 30 hota hai.
Sabse chhota 4-digit number = 1000
1000 ÷ 30 = 33.33...
Next multiple = 34 × 30 = 1020
Answer: 1020
Important Equations — Ek Nazar Me
| Concept | Formula / Rule |
|---|---|
| Divisibility by 2 | Last digit 0, 2, 4, 6, ya 8 ho |
| Divisibility by 3 | Digits ka sum 3 se divisible ho |
| Divisibility by 9 | Digits ka sum 9 se divisible ho |
| Divisibility by 11 | (Odd place digits ka sum) − (Even place digits ka sum) = 0 ya 11 ka multiple |
| HCF × LCM relation | HCF(a, b) × LCM(a, b) = a × b |
| Euclid's Division Lemma | a = bq + r, 0 ≤ r < b |
| Baudhayana–Pythagoras theorem | Hypotenuse² = Base² + Height² |
| Pythagorean triplet generator | (2m, m² − 1, m² + 1) jahan m > 1 |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Divisibility by 11 ka rule apply karte waqt odd-even places ka order ulta le lena — hamesha right se count karo, left se nahi.
- HCF aur LCM ka formula (HCF × LCM = a × b) sirf do numbers ke liye directly kaam karta hai, teen ya zyada numbers ke liye ye seedha apply nahi hota.
- Euclid's Division Lemma mein bada number (dividend) ko galat position pe rakh dena — hamesha bade number ko 'a' aur chhote ko 'b' maano, warna steps ulte ho jaate hain.
- Prime factorisation karte waqt 1 ko prime number maan lena — 1 na prime hai na composite, ye ek common conceptual galti hai.
- Pythagorean triplet generate karte waqt formula (2m, m²−1, m²+1) mein m ki value 1 le lena, jisse ek term zero ban jaata hai — hamesha m > 1 lena chahiye.
- Co-prime numbers ka matlab 'dono prime hone chahiye' samajh lena — jabki co-prime ka matlab sirf itna hai ki unka HCF 1 ho, dono numbers khud prime hone zaroori nahi (jaise 8 aur 9).
Board-Style Important Questions
- Board-style question: HCF aur LCM ka relation use karke do numbers ka LCM find karna, jab unka HCF aur product diya ho.
- Euclid's Division Algorithm ka use karke do given numbers ka HCF step-by-step nikalna.
- Divisibility rules (2, 3, 9, 11) apply karke multi-digit numbers ko test karna — ye conceptual application-based questions frequently exam patterns mein dikhte hain.
- Pythagorean triplet find karna jab ek number diya ho — geometry aur number theory ka combined application.
- Ancient Indian mathematics (Baudhayana Sulba Sutra) ke context mein short-answer conceptual question — historical significance aur modern connection explain karna.
- Prime factorisation method se HCF-LCM dono ek saath nikalna — factor tree diagram ke saath present karna.
Aksar Poochhe Jaane Wale Sawaal
NCERT Class 8 Maths ka naya book naam kya hai 2026-27 session ke liye?
2026-27 session ke liye NCERT ne Class 8 Maths ko purani single-book format se hata kar do-part 'Ganita Prakash' series mein rationalise kar diya hai — Part 1 (Chapters 1–7) aur Part 2 (Chapters 8–14), total 14 chapters.
Chapter 3 'A Story of Numbers' mein kya-kya topics cover hote hain?
Is chapter mein divisibility rules, factors aur multiples, prime factorisation, HCF/LCM, Euclid's division lemma, aur numbers ki historical journey — jisme Baudhayana Sulba Sutra jaisa ancient Indian contribution bhi included hai — cover kiya jaata hai.
Ganita Prakash Part 1 aur Part 2 mein kitne chapters hain?
Part 1 mein Chapters 1 se 7 tak hain, aur Part 2 mein Chapters 8 se 14 tak — total milakar 14 chapters poore Class 8 Maths syllabus ko cover karte hain.
Purane NCERT Class 8 Maths book aur naye Ganita Prakash mein kya farak hai?
Sabse bada farak format ka hai — single book se do-part structure. Content presentation bhi zyada activity-based aur conceptual storytelling wali ho gayi hai, jaise chapter names 'A Story of Numbers', 'Tales by Dots and Lines' isse reflect karte hain. Exact drop/merge hue purane topics ka poora list official ncert.nic.in rationalisation PDF se verify nahi ho paya is session mein, isliye woh claim yahan nahi banayi gayi.
Euclid's Division Lemma aur Division Algorithm mein kya farak hai?
Euclid's Division Lemma ek single statement hai — 'a = bq + r, jahan 0 ≤ r < b' — jo do numbers ke division ko define karta hai. Euclid's Division Algorithm is lemma ko baar-baar (repeatedly) apply karne ka process hai jab tak remainder 0 na ho jaaye, taaki HCF nikala ja sake.
Baudhayana ka Pythagoras theorem se kya connection hai, jo is chapter mein padhaya jaata hai?
Baudhayana Sulba Sutra mein right-angled triangle ka wahi rule (hypotenuse² = base² + height²) likha mil chuka tha jo baad mein Greek mathematician Pythagoras ke naam se famous hua. Isi wajah se NCERT ise 'Baudhayana–Pythagoras theorem' ke context mein introduce karta hai, taaki students ko iski Indian origin bhi pata chale.
Class 8 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.