Class 8 Maths · Chapter 5
Short answer:
Quick Summary: Chapter 5 "Number Play" (Ganita Prakash Part 1) numbers ke patterns explore karta hai — palindrome numbers, Kaprekar numbers, Kaprekar constant (6174), Collatz conjecture, aur Armstrong numbers. Iska focus rote memorization nahi balki reasoning aur pattern-discovery hai.
Ganita Prakash Class 8 Maths Chapter 5, "Number Play", ek bahut interesting chapter hai jo numbers ki duniya me chhupe patterns aur puzzles ko explore karta hai — palindromes, Kaprekar numbers, Kaprekar constant 6174, aur Collatz conjecture jaisi mazedaar cheezein. NCERT ne Class 8 Maths ko purani single-book format se hata kar do-part Ganita Prakash (Part 1 + Part 2) me rationalise kiya hai 2026-27 session ke liye — total 14 chapters, jisme Part 1 me Chapters 1-7 aur Part 2 me Chapters 8-14 aate hain. Number Play, Part 1 ka Chapter 5 hai.
Ye chapter students ko yaad rakhwane ke bajaye khud explore karne pe focus karta hai — numbers ke saath khelte hue logical reasoning, pattern-recognition, aur mathematical curiosity develop hoti hai. Iske alawa Ganita Prakash ke doosre chapters jaise A Square and A Cube, Baudhayana aur Pythagoras Theorem, Proportional Reasoning 1 vs 2, aur We Distribute Yet Things Multiply bhi is naye curriculum ka hissa hain jo ek connected, activity-based learning journey banate hain.
Chapter 5 Summary — 5 Minute Revision
"Number Play" Class 8 Ganita Prakash ka woh chapter hai jo numbers ko sirf calculation ka tool nahi, balki exploration aur discovery ka zariya banata hai. Palindromes se lekar Kaprekar constant 6174 tak, aur Collatz conjecture ki unsolved mystery tak — har concept students ko step-by-step working ke through mathematical thinking sikhata hai.
Is chapter ki practice karte waqt Kaprekar routine ke steps carefully likhna, Collatz sequence me even/odd rule sahi apply karna, aur palindrome/Armstrong number checks accurately karna zaroori hai — jaise koi bhi range me smallest ya largest number dhoondte waqt har candidate ko systematically check karna, koi bhi valid case skip nahi karna. Agla chapter "Tales by Dots and Lines" graph-based concepts introduce karega, jabki Part 2 ke chapters jaise Fractions in Disguise, Algebra Play, aur Exploring Some Geometric Themes is foundation ko aage le jaate hain. Ganita Prakash ka poora structure old NCERT book se kaafi different hai — activity-based aur discovery-driven approach naya highlight hai.
In-Text Questions — Solutions
Figure it Out (Page activity) — 2, 3, 4 aur 5 anko wale kitne palindrome numbers ban sakte hain jinme sirf digits 1 aur 2 ka use ho?
2-digit: dono digits same hone chahiye → 11, 22 → 2 numbers.
3-digit: pehla aur teesra digit same hona chahiye, beech ka digit bhi 1 ya 2 me se koi ek → 2 (first-last choice) × 2 (middle choice) = 4 numbers: 111, 121, 212, 222.
4-digit: pehle do digits fix karo, aakhri do unka mirror image ban jaate hain → 2 × 2 = 4 numbers: 1111, 1221, 2112, 2222.
Pattern: har case me 2 choices har 'free' position ke liye milti hain, isliye count = 2(number of free positions) hota hai.
Kya 10 se 99 ke beech koi aisa 2-digit number hai jiska square palindrome ho?
Check karte hain kuch numbers:
112 = 121 (palindrome!)
222 = 484 (palindrome!)
262 = 676 (palindrome!)
Haan, 11, 22, 26 jaise kai numbers hain jinka square palindrome hai.
Kaprekar routine 4-digit number 1000 (jisme sab digits same nahi hote ideally) par kyun kaam nahi karega?
Kaprekar routine tabhi kaam karta hai jab number ke sabhi digits same na ho.
1000: digits descending = 1000, ascending = 0001
1000 − 0001 = 0999 → yahan se aage process chalega, lekin agar sab digits same ho (jaise 1111), to descending − ascending = 0 ho jaata hai aur process ruk jaata hai.
Isliye rule hai: sab digits same wale numbers is routine ke liye exclude kiye jaate hain.
Exercise Questions — Solutions (Q1–Q6)
Q1. 100 se 200 ke beech ka sabse chhota palindrome number likho jo 3 se divisible ho.
100 se 200 ke beech palindrome numbers: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191.
3 se divisible check karne ke liye digit-sum lo:
101 → 1+0+1 = 2 (na), 111 → 1+1+1 = 3 (haan!)
Answer: 111 (1+1+1 = 3, jo 3 se divisible hai, aur 100-200 ke beech ye sabse chhota aisa palindrome hai)
Q2. Ek Kaprekar number kya hota hai? 45 ko lekar Kaprekar routine ka ek step dikhao.
Kaprekar number woh hota hai jiska square do parts me todne par un parts ka sum wapas woh number bana de.
452 = 2025
2025 ko split karo: 20 aur 25
20 + 25 = 45
Haan, 45 ek Kaprekar number hai kyunki split karke add karne par original number wapas mil raha hai.
Q3. Kaprekar constant 6174 nikalne ke liye 3524 se shuru karke steps dikhao.
Rule: digits ko descending aur ascending order me arrange karo, phir subtract karo.
Step 1: 3524 → 5432 − 2345 = 3087
Step 2: 3087 → 8730 − 0378 = 8352
Step 3: 8352 → 8532 − 2358 = 6174
3 steps me 6174 (Kaprekar constant) mil gaya, jo aage repeat hota rahega.
Q4. Collatz conjecture ke rule follow karke 7 se shuru karke sequence likho jab tak 1 na aaye.
Rule: agar number even hai to 2 se divide karo, odd hai to (3n+1) karo.
7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Total 16 steps me 1 tak pahunche — yahi Collatz conjecture ka observation hai ki har positive integer aakhir me 1 pe pahunch jaata hai.
Q5. Kya 1 se 100 tak koi bhi number aisa hai jiske digits ka cube-sum wapas usi number ke barabar ho (Armstrong number)? Ek example do.
Armstrong number woh hota hai jiske digits ke cubes ka sum khud number ke barabar ho.
153: 13 + 53 + 33 = 1 + 125 + 27 = 153
153 ek Armstrong number hai (3-digit range me), jo dikhata hai numbers me chhupe patterns kitne interesting ho sakte hain.
Q6. Ek 3-digit number socho jiske digits ka product 2-digit number aata hai. Us number ko explain karo pattern ke saath.
Example: number 439 lo.
Digit product = 4 × 3 × 9 = 108 (3-digit ban gaya, isliye try karo dusra)
Number 234 lo: 2 × 3 × 4 = 24 (2-digit)
234 ek valid example hai jiska digit-product 24 hai — 2-digit number.
Important Equations — Ek Nazar Me
| Concept | Rule / Formula | Example |
|---|---|---|
| Palindrome Number | Aage se ya peeche se same digits | 121, 3553 |
| Kaprekar Number | n2 ko split karo → parts add karne par n wapas mile | 452=2025 → 20+25=45 |
| Kaprekar Constant Routine | Digits descending − ascending order, repeat | 3524 → ... → 6174 |
| Collatz Rule | Even: n/2 | Odd: 3n+1 | 7 → 22 → 11 → ... → 1 |
| Armstrong Number | Digits ke cubes ka sum = number khud | 153 = 13+53+33 |
| Digit Sum Divisibility (by 3) | Digit sum agar 3 se divisible ho, to number bhi hoga | 111 → 1+1+1=3 (÷3) |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Kaprekar routine me digits ko galat order me arrange karna — hamesha descending minus ascending hi karna hai, ulta nahi.
- Collatz sequence me even/odd rule mix up karna — even ko 3n+1 se treat kar dena, jo galat hai.
- Palindrome number check karte waqt kisi bhi valid palindrome ko chhod dena — jaise 100-200 range me 111 ko miss karke seedha 121 se list start kar dena; digit-sum check har candidate par lagana zaroori hai, chhote se chhota number pehle verify karo.
- Armstrong number me power (cube vs square) galat use karna — 3-digit number ke liye hamesha cube hi use hota hai, square nahi.
- Kaprekar constant 6174 tak pahunchne ke steps count karte waqt beech ka koi step skip kar dena — har step likhna zaroori hai working dikhane ke liye.
- Digit-sum divisibility rule sirf 3 aur 9 ke liye hi apply karna — ise galti se 4, 6 jaise numbers pe bhi apply kar dena, jo sahi nahi hai.
Board-Style Important Questions
- Ganita Prakash naye syllabus ke saath 2026-27 se pehle NCERT exam papers is naye chapter format me available nahi hain, isliye specific PYQ years yahan claim nahi kiye ja rahe.
- School-level periodic tests me is chapter se Kaprekar routine aur palindrome-based reasoning questions frequently practice karwaye jaate hain.
- Class 8 board-pattern assessments (jahan school apna paper banata hai) me digit-pattern aur number-property based short-answer questions common trend rahe hain purane 'Playing with Numbers' chapter se bhi.
- CBSE ka koi centralized Class 8 board exam nahi hota, isliye school-specific periodic assessments hi is chapter ka main testing ground hain.
- Practice ke liye Kaprekar constant, Collatz sequence, aur Armstrong/palindrome numbers par based reasoning questions sabse zyada expected pattern follow karte hain.
Aksar Poochhe Jaane Wale Sawaal
Ganita Prakash Class 8 Chapter 5 'Number Play' kya sikhata hai?
Ye chapter number patterns, palindromes, Kaprekar numbers, Kaprekar constant (6174), Collatz conjecture, aur digit-based puzzles explore karta hai. Isme rote formulas nahi, balki numbers ke saath khelte hue logical reasoning develop ki jaati hai.
Kya Number Play chapter exam ke liye important hai?
Ye chapter conceptual aur activity-based hai — direct formula-based questions kam aate hain, lekin reasoning-based aur pattern-identification wale questions poochhe ja sakte hain. Practice zaroor karo taaki number properties clear rahein.
6174 (Kaprekar constant) kaise nikalte hain?
Kisi bhi 4-digit number (jisme sab digits same na ho) ko lo, digits ko descending aur ascending order me arrange karo, badi se choti minus karo. Ye process repeat karne par max 7 steps me 6174 par aa jaata hai.
Palindrome number kya hota hai?
Palindrome number woh hota hai jo aage se ya peeche se same padhta hai — jaise 121, 3553, 45654. Chapter 5 me palindrome patterns aur unke sums explore kiye gaye hain.
Ganita Prakash Part 1 aur Part 2 me kitne chapters hain?
Ganita Prakash Class 8 ko do parts me rationalise kiya gaya hai — Part 1 me Chapters 1-7 aur Part 2 me Chapters 8-14, total 14 chapters, 2026-27 session ke liye.
Collatz conjecture kya hai aur ye chapter me kyun important hai?
Collatz conjecture kehta hai ki koi bhi positive number lo — even hai to 2 se divide karo, odd hai to 3n+1 karo — end me hamesha 1 par pahunch jaate ho. Ye ek unsolved math problem hai jo isme intuition build karne ke liye diya gaya hai.
Class 8 Maths — Saare Chapters
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