✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 7 × 7 = ?
- 12 ke saare factors likho.
- Ek square ki side 6 cm hai. Area?
Answers dekho
- 49
- 1, 2, 3, 4, 6, 12 (chhe factors)
- 36 sq cm
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, multiplication tables aur factors ek baar dohra lijiye, phir yeh post bilkul aasaan lagega.
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🎬 Kahani: Rani Ratnamanjuri Ke 100 Lockers
Rani Ratnamanjuri ne apne mahal mein 100 lockers chhode, sab band. 100 log ek-ek karke aate hain: Person 1 har locker kholta hai, Person 2 har 2nd locker ko palat-ta hai (khula ho to band, band ho to khula), Person 3 har 3rd locker ko… aise hi Person 100 tak. 🔐
Aakhir mein sirf kuch lockers khule rehte hain, aur unke number milke khazane ka passcode banate hain. Khoisnam ne bina ek bhi locker chhuye bata diya ki kaunse khule rahenge! 🤯 Kaise? Aaj yahi raaz kholenge.
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📖 Concept: Number × Khud = Square Number
💡 Is post ki ek badi baat: Square number woh hai jo kisi number ko khud se guna karke bane: n × n = n². Iski jaan uske factors ki jodi mein hai: har factor ka partner hota hai, sirf square mein ek factor (√n) apna hi partner hota hai, isliye uske factors ki ginti odd hoti hai.
Step 1: Square ka matlab: tiles ka square
Agar 1 × 1 ki tiles se poora square bharna ho to kitni tiles lagengi? Side 1 → 1 tile, side 2 → 4, side 3 → 9, side 4 → 16, side 5 → 25.
n × n = n² (padhte hain “n ka square”)
WHY “square”? Kyunki n² ek square ka area hai jiski side n hai. Naam hi shakal se aaya hai.
Step 2: Perfect square
Natural numbers (1, 2, 3, …) ke squares ko perfect squares kehte hain: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
Kya 32 perfect square hai? 5² = 25, 6² = 36. 32 beech mein hai, kisi natural number ka square nahi. Nahi.
Step 3: Lockers kitni baar palte?
Locker #6 ko kaun chhoota hai? Jinka number 6 ko poora divide kare: Person 1, 2, 3, 6. Yaani 6 ke factors! 4 baar palta → band (khula → band → khula → band).
Niyam: factors ki ginti odd → locker khula. Even → band.
Step 4: Factors jodi mein kyun aate?
6 = 1 × 6 = 2 × 3. Har factor ka ek partner: (1, 6), (2, 3). Jodiyan → ginti even.
36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6. Yahan 6 ka partner khud 6! Usse ek hi baar ginte hain → factors 1, 2, 3, 4, 6, 9, 12, 18, 36 = 9 (odd).
WHY yeh kaam karta? Sirf square number mein hi koi factor “khud se guna” hoke number banata hai. Isliye sirf squares ke factors odd.
Step 5: Paheli ka jawab
Khule lockers = perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (10 lockers). Passcode ke pehle 5: 1, 4, 9, 16, 25.
Check: locker 16 ke factors 1, 2, 4, 8, 16 = 5 (odd) ✅ khula; locker 15 ke 1, 3, 5, 15 = 4 (even) ✅ band.
🪜 Atak Gaye? Aise Socho
- Number ko do barabar numbers ke guna mein todne ki koshish karo.
- Table yaad nahi? n = 1, 2, 3… ka n × n likhte jao jab tak number paar na ho.
- Number do squares ke beech aaya? To perfect square nahi.
- Factors ginne hain? Jodiyon mein likho: (1, n), (2, n/2)…
- Koi jodi mein dono same? To square hai.
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🔐 Deep Concept: Sirf Do Baar Palte Lockers Kaun? (Prime Numbers Ka Raaz)
NCERT ka agla twist: kaunse lockers sirf do baar palte? Jinke sirf 2 factors hon: 1 aur khud. Yaani prime numbers: 2, 3, 5, 7, 11, … Prime ke factors (1, p) ek hi jodi, isliye kabhi khule nahi rehte.
Teen baar kaun palta? Jinke 3 factors: 4 (1, 2, 4), 9 (1, 3, 9), 25, 49. Yeh prime ke squares hain! Kyunki p² ke factors sirf 1, p, p².
Gehri baat: factors ki ginti number ki “banawat” batati hai. Even ginti = sab factors jodi mein. Odd ginti = ek akela factor beech mein khada, aur woh factor hi square root hai. Yahi soch post 4 (square root) aur post 7 (cube root) tak jayegi. 🎯
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✏️ Solved Examples
Example 1 (Easy): Kya 49 perfect square hai? Factors ginke batao.
49 = 7 × 7 → haan, 7². Factors: 1, 7, 49 = 3 (odd).
Check: odd factors = square ✅
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Example 2 (Medium): Locker #24 aakhir mein khula ya band?
Jodiyan: (1, 24), (2, 12), (3, 8), (4, 6) → 8 factors, even → band.
Check: 24 do squares 16 aur 25 ke beech, square nahi, isliye band ✅
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Example 3 (NCERT style): NCERT style: 1 se 50 tak lockers mein kitne khule rahenge?
Perfect squares ≤ 50: 1, 4, 9, 16, 25, 36, 49 → 7.
Check: 7² = 49 ≤ 50 aur 8² = 64 > 50, to 7 hi ✅
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Khoisnam ka raaz: “Lockers ko palatna = factors ginna. Aur odd factors sirf squares ke hote hain!” 🔓
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🏠 Square Numbers Roz Ki Zindagi Mein
- 🏠 Farsh ki tiles: 1 ft ki tiles se 12 ft × 12 ft ka kamra = 144 tiles.
- ♟️ Chessboard: 8 × 8 = 64 khaane.
- 🌱 Bagicha: 10 qatar × 10 paudhe = 100 paudhe, square khet.
- 🏫 PT drill: 49 bachche 7 × 7 ke square mein khade ho sakte hain, 50 nahi.
- 📐 Area: zameen “square metre” mein naapi jaati hai, square ka area hi.
Jab bhi “barabar qatar, barabar column” dikhe, square number chhupa hai.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Number = n × n check karna | ✅ Squares ki list se milao | Seedha aur tez |
| Bada number (jaise 1156) | ✅ Prime factorisation (Post 4) | List lambi ho jaati |
| Factors ki ginti odd/even | ✅ Jodiyon mein likho | Square mein ek akela factor |
| Number do squares ke beech | ❌ Perfect square nahi | Beech mein koi n² nahi |
| n² ko 2 × n samajhna | ❌ Kabhi nahi | Square = n × n, n + n nahi |
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❌→✅ Worked Mistake
Sawaal: 6² kitna?
❌ Galat: “6² = 6 × 2 = 12.”
✅ Sahi: chhota 2 batata hai kitni baar khud se guna: 6² = 6 × 6 = 36. Check: 6 cm side ke square mein 6 qatar × 6 tiles = 36 ✅
🧠 Rule of thumb: Square = number × wahi number; chhota 2 “do baar” hai, “guna 2” nahi.
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📝 10 MCQs
Q1. 8² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 8 × 8 = 64, sahi jawab ✅
- (b) galat: 16 to 8 × 2 hai, square nahi.
- (c) galat: 10 to 8 + 2 hai, galat soch.
- (d) galat: 82 sirf digits jodke likha, galat.
Q2. Inmein perfect square kaunsa?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 81 = 9 × 9, isliye perfect square ✅
- (b) galat: 80 do squares 64 aur 81 ke beech hai.
- (c) galat: 82 do squares 81 aur 100 ke beech hai.
- (d) galat: 90 bhi 81 aur 100 ke beech, square nahi.
Q3. 36 ke kitne factors?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 1, 2, 3, 4, 6, 9, 12, 18, 36 = 9 ✅
- (b) galat: 8 tab hote jab 6 ka partner alag hota.
- (c) galat: 6 bahut kam hai, kai factors chhoote.
- (d) galat: 12 zyada hai, kuch factors do baar gine.
Q4. 100 lockers mein kitne khule rahenge?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 1² se 10² tak, 10 squares ✅
- (b) galat: 50 to even lockers hain, galat soch.
- (c) galat: 25 ka koi aadhar nahi, galat.
- (d) galat: 100 = 10² bhi khula, isliye 10 hoga.
Q5. Locker #12 aakhir mein?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 12 ke 6 factors, even, isliye band ✅
- (b) galat: Khula tab jab factors odd hon.
- (c) galat: Do baar khula koi state nahi hai.
- (d) galat: Paheli mein locker toot-ta nahi hai.
Q6. Prime number wala locker kitni baar palta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Factors sirf 1 aur khud, do baar ✅
- (b) galat: Ek baar sirf locker #1 palta hai.
- (c) galat: Teen baar prime ke square palte hain.
- (d) galat: Person 1 to har locker chhoota hai.
Q7. Kis number ke exactly 3 factors?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 25 = 5², factors 1, 5, 25 ✅
- (b) galat: 12 ke 6 factors hain, isliye galat.
- (c) galat: 15 ke 4 factors: 1, 3, 5, 15.
- (d) galat: 16 ke 5 factors: 1, 2, 4, 8, 16.
Q8. Side 9 cm wale square ka area?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 9 × 9 = 81 sq cm ✅
- (b) galat: 18 to 9 × 2 hai, galat soch.
- (c) galat: 36 perimeter hai, area nahi.
- (d) galat: 90 ka koi aadhar nahi, galat.
Q9. 50 bachche poore square mein khade ho sakte?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 50 perfect square nahi, isliye nahi ✅
- (b) galat: 7 × 7 = 49 hi hote, 50 nahi.
- (c) galat: 5 × 10 rectangle hai, square nahi.
- (d) galat: 25 × 2 bhi rectangle hai, square nahi.
Q10. Square ke factors odd kyun?
Solution dekho
Sahi jawab: (a)
- (a) sahi: √n ka partner khud √n, ek baar gina ✅
- (b) galat: 36 ke factors mein 2, 4 even bhi hain.
- (c) galat: Bada hona ginti nahi badalta.
- (d) galat: Kaaran saaf hai: jodi ka niyam.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. n² aur 2n mein kya farak?
n² = n × n (guna), 2n = n + n (jod). 5² = 25, 2 × 5 = 10.
2. Kya 0 ka square hota?
Haan, 0 × 0 = 0. Par perfect squares natural numbers (1, 2, 3…) se bante hain.
3. Kya 1 perfect square hai?
Haan, 1 = 1 × 1. Iska ek hi factor (1), jo odd hai ✅.
4. Fraction ka square?
Haan: (2/3)² = 4/9. Upar-neeche dono ka square.
5. Negative ka square?
(−6)² = 36, positive. Minus × minus = plus.
6. Aage kya?
Squares ke aakhri digit ke niyam: Post 2 — Square Numbers Ki Properties.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 11² = ?
Solution dekho
121. Check: 11 × 11 = 110 + 11 ✅
Q2. Easy Kya 64 perfect square?
Solution dekho
Haan, 8². Check ✅
Q3. Easy 1 se 20 tak perfect squares?
Solution dekho
1, 4, 9, 16. Check: 5² = 25 > 20 ✅
Q4. Easy 15² = ?
Solution dekho
225. Check: 15 × 10 + 15 × 5 = 150 + 75 ✅
Q5. Medium Locker #49: khula ya band?
Solution dekho
Factors 1, 7, 49 (odd) → khula. Check ✅
Q6. Medium 18 ke factors ginke batao, square hai?
Solution dekho
(1,18), (2,9), (3,6) → 6, even → nahi. Check ✅
Q7. Medium Kamra 14 × 14 ft, 1 ft tiles?
Solution dekho
196. Check: 14 × 14 = 140 + 56 ✅
Q8. Medium 30 aur 50 ke beech perfect squares?
Solution dekho
36, 49. Check: 5² = 25, 8² = 64 ✅
Q9. NCERT style Do baar palne wale pehle 5 lockers?
Solution dekho
Primes: 2, 3, 5, 7, 11. WHY: sirf do factors.
Q10. NCERT style 1 se 100 mein teen baar palne wale lockers?
Solution dekho
Prime ke squares: 4, 9, 25, 49. Check: 11² = 121 > 100 ✅
Q11. NCERT style 1 se 200 lockers mein kitne khule?
Solution dekho
14² = 196 ≤ 200, 15² = 225 → 14. Check ✅
Q12. NCERT style Kya 2032 perfect square?
Solution dekho
45² = 2025, 46² = 2116 → nahi. Check ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Ek locker 5 baar palta. Uska number 1 se 100 mein kya ho sakta?
Solution dekho
5 factors = p⁴: 16, 81. Check: 16 ke 1, 2, 4, 8, 16 ✅
Q2. Kya kisi square ke factors even ho sakte?
Solution dekho
Nahi, √n hamesha akela. Check: 49 → 3 factors ✅
Q3. 100 ki jagah 1000 lockers: kitne khule?
Solution dekho
31² = 961, 32² = 1024 → 31. Check ✅
Q4. Kaunse lockers sirf ek baar palte?
Solution dekho
Sirf #1 (ek hi factor). Check ✅
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🏆 Olympiad Corner
Q1. 1 se 100 mein sabse zyada factors wala square?
Solution dekho
36 (9 factors); 100 ke bhi 9.
Q2. Perfect squares 1 se 1000 tak kitne?
Solution dekho
31.
Q3. n² − n hamesha even kyun?
Solution dekho
n(n − 1): do lagataar numbers, ek even.
Q4. Khule lockers ke numbers ka jod (1 se 100)?
Solution dekho
1 + 4 + … + 100 = 385.
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💗 CharuMam Motivation
Ek paheli, aur poora chapter khul gaya: “square ke factors odd hote hain.” Maths ratna nahi, patterns dhoondhna hai. Aap bhi Khoisnam ban sakte ho! 🌟
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🟢 Master Recap
- 💡 Chaabi: Square number woh hai jo kisi number ko khud se guna karke bane: n × n = n². Iski jaan uske factors ki jodi mein hai: har factor ka partner hota hai, sirf square mein ek factor (√n) apna hi partner hota hai, isliye uske factors ki ginti odd hoti hai.
- n² = n × n, square ka area.
- Perfect squares: 1, 4, 9, 16, 25, …
- Factors jodi mein aate hain.
- Square mein √n akela → factors odd.
- Prime: 2 factors; prime²: 3 factors.
- Khule lockers = perfect squares.
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👉 Aage Kya?
Ab squares ke aakhri digit aur zero ke niyam dekhenge: Post 2 — Square Numbers Ki Properties.
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