✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 4 × 4 × 4 = ?
- 5² = ?
- (−2) × (−2) = ?
Answers dekho
- 64
- 25
- 4
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Square Numbers Kya Hain? ka n² wala matlab dobara dekh lijiye.
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🎬 Kahani: Hospital Ki Taxi 1729
1918 mein mathematician G. H. Hardy, beemar Srinivasa Ramanujan se milne hospital gaye. Bole: “Meri taxi ka number 1729 tha, bada boring number.” 🚕
Ramanujan turant bole: “Nahi! Yeh to bahut dilchasp hai. Yeh sabse chhota number hai jo do cubes ke jod se do alag tareeke se banta hai!” Cube kya hota hai, aur 1729 mein aisa kya khaas? Chaliye dekhte hain.
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📖 Concept: Number × Khud × Khud = Cube
💡 Is post ki ek badi baat: Cube number = kisi number ko teen baar khud se guna: n³ = n × n × n, jo n side wale ghanaakaar (cube) ka volume hai. Square mein cheezein jodi mein aati thi, cube mein teen-teen mein: zero bhi 3, 6, 9… mein, aur prime factors bhi triplets mein.
Step 1: Cube ka matlab: chhote cubes ka dher
4 side wala cube: har layer mein 4 × 4 = 16 chhote cube, aur aisi 4 layers → 4 × 4 × 4 = 64.
n × n × n = n³ (padhte “n ka cube”)
WHY “cube”? Kyunki n³ ek cube ka volume hai. Perfect cubes: 1, 8, 27, 64, 125, …
Step 2: 1 se 20 tak cubes
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 11 | 1331 |
| 2 | 8 | 12 | 1728 |
| 3 | 27 | 13 | 2197 |
| 4 | 64 | 14 | 2744 |
| 5 | 125 | 15 | 3375 |
| 6 | 216 | 16 | 4096 |
| 7 | 343 | 17 | 4913 |
| 8 | 512 | 18 | 5832 |
| 9 | 729 | 19 | 6859 |
| 10 | 1000 | 20 | 8000 |
9 cube hai? 2³ = 8, 3³ = 27 → 9 (aur 10 se 26) cube nahi.
Step 3: Aakhri digit: sab possible!
Square sirf 0, 1, 4, 5, 6, 9 pe khatam hote, par cube har digit pe ho sakte: 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0.
WHY kaam ka? Har aakhri digit ka sirf ek root digit! Isliye cube root ka guess bahut aasaan (Post 7).
Step 4: Zero teen-teen mein, aur negative
10³ = 1000, 100³ = 10,00,000. Zero 3, 6, 9… → cube exactly 00 (do zero) pe khatam nahi ho sakta.
(−6)³ = −6 × −6 × −6 = −216. Square hamesha positive, par negative ka cube negative. Fraction: (4/6)³ = 64/216.
Step 5: Taxicab numbers
1729 = 1³ + 12³ = 1 + 1728, aur 1729 = 9³ + 10³ = 729 + 1000. Do alag tareeke!
Aise numbers taxicab numbers kehlaate. Agle do: 4104 = 2³ + 16³ = 9³ + 15³, aur 13832 = 2³ + 24³ = 18³ + 20³.
🪜 Atak Gaye? Aise Socho
- Cube = teen baar guna, “× 3” nahi.
- Pehle n², phir usko n se guna: 7² = 49, 49 × 7 = 343.
- Table ke pehle 10 cubes yaad karo.
- Number do cubes ke beech? To cube nahi.
- Negative ka cube: sign minus hi rahega.
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🔐 Deep Concept: Cubes Kitne Door-Door Hote Hain? (Digit Ginti)
1-digit cubes: 1, 8 (2). 2-digit: 27, 64 (2). 3-digit: 125 se 729 (5). 4-digit: 1000 se 9261 (10³ se 21³, 12).
Squares ke mukable cubes bahut tezi se badhte, isliye bahut kam aur dur-dur. 2-digit number ka cube kam se kam 1000 (4 digit) aur zyada se zyada 99³ = 9,70,299 (6 digit).
Gehri baat: square = 2D (area), cube = 3D (volume). Side 2 guna → area 4 guna, volume 8 guna. Yahi kaaran hai ki bada matka ya tank thoda bada dikhe par paani bahut zyada aata! 🎯
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✏️ Solved Examples
Example 1 (Easy): 7³ = ?
7 × 7 = 49, 49 × 7 = 343.
Check: aakhri digit 7 → 3 ✅
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Example 2 (Medium): Kya 100 perfect cube?
4³ = 64, 5³ = 125 → nahi.
Check: exactly 2 zero, cube mein zero teen-teen ✅
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Example 3 (NCERT style): NCERT style: 4104 ko do cubes ke jod ke do tareeke se dikhao.
2³ + 16³ = 8 + 4096 = 4104; 9³ + 15³ = 729 + 3375 = 4104.
Check: dono jod 4104 ✅
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Ramanujan ka jaadu: “Har number ek dost hai.” 1729 ab boring nahi! 🚕✨
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🏠 Cubes Roz Ki Zindagi Mein
- 🧊 Ice cubes / dice: sab sides barabar.
- 🎲 Rubik cube: 3 × 3 × 3 = 27 chhote tukde.
- 📦 Dabba packing: 1 cm cubes se 10 cm ka dabba = 1000 cubes.
- 💧 Paani ki tanki: 1 m³ = 1000 litre.
- 🧱 Eenton ka dher: 5 × 5 × 5 = 125 eent.
Lambaai, chaudaai, oonchaai teeno barabar → cube.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Volume of cube | ✅ side³ | Teen dimension |
| n³ ko 3n samajhna | ❌ Kabhi nahi | 2³ = 8, 6 nahi |
| Aakhri digit se cube check | ⚠️ Har digit possible | Square jaisa rad nahi hota |
| Zero ginti 1, 2, 4, 5… | ✅ Cube nahi | Zero teen-teen mein |
| Negative ka cube | ✅ Negative | Teen minus = minus |
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❌→✅ Worked Mistake
Sawaal: 5³ = ?
❌ Galat: “5 × 3 = 15.”
✅ Sahi: 5 × 5 × 5 = 125. Check: 5 side ke cube mein 5 layers × 25 = 125 ✅
🧠 Rule of thumb: Cube = teen baar wahi number; chhota 3 “teen baar” hai, “guna 3” nahi.
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📝 10 MCQs
Q1. 6³ = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 6 × 6 × 6 = 216 ✅
- (b) galat: 18 to 6 × 3 hai, galat.
- (c) galat: 36 to 6² hai, cube nahi.
- (d) galat: Yeh to bilkul alag cheez hai.
Q2. Kaunsa perfect cube?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Bilkul sahi jawab: 8³ = 512 ✅
- (b) galat: 500 = 2² × 5³, 2 adhoora.
- (c) galat: 256 = 2⁸, 8 teen se nahi bata.
- (d) galat: 400 7³ aur 8³ ke beech hai.
Q3. (−3)³ = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Teen minus = minus, −27 ✅
- (b) galat: Sign minus rehna chahiye.
- (c) galat: −9 to sirf do baar guna.
- (d) galat: 9 to square hai, cube nahi.
Q4. 1729 = 9³ + ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 729 + 1000 = 1729 ✅
- (b) galat: 12³ to 1³ ke saath jodi hai.
- (c) galat: 11³ = 1331, jod 2060, isliye yeh galat.
- (d) galat: 1³ to 12³ ke saath chalta.
Q5. Cube kis digit pe khatam ho sakta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 0 se 9 sab aate ✅
- (b) galat: Yeh to squares ka niyam hai.
- (c) galat: 8 (2³) even pe khatam hai.
- (d) galat: 27 odd pe khatam hai, isliye yeh galat.
Q6. Kya cube exactly 00 pe khatam?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Zero teen-teen mein aate ✅
- (b) galat: 10³ = 1000 mein 3 zero.
- (c) galat: Niyam pakka, kabhi kabhi nahi.
- (d) galat: Bada hona ginti nahi badalta.
Q7. Side 2 guna, volume?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 2 × 2 × 2 = 8 ✅
- (b) galat: Sirf ek taraf double se.
- (c) galat: 4 guna area hota hai, isliye yeh galat.
- (d) galat: 6 sides ki ginti hai, isliye yeh galat.
Q8. 3-digit cubes kitne?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 125 se 729 tak 5 ✅
- (b) galat: 2 to 2-digit cubes hain.
- (c) galat: 9 tak ginti galat hai.
- (d) galat: 1000 4-digit hai, isliye yeh galat.
Q9. 12³ = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 144 × 12 = 1728 ✅
- (b) galat: 144 to 12² hai, isliye yeh galat.
- (c) galat: 1331 to 11³ hai, isliye yeh galat.
- (d) galat: 2197 to 13³ hai, isliye yeh galat.
Q10. Taxicab number ka matlab?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 1729 jaisa, do alag jodi ✅
- (b) galat: 1729 khud cube nahi hai.
- (c) galat: Naam kahani se aaya bas.
- (d) galat: Dono cubes hi hote hain.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. n³ ka padhna?
“n cube” ya “n ki power 3”.
2. Sanskrit mein cube?
Ghana (Post 8 mein itihaas).
3. Kya 0 cube?
0³ = 0, haan.
4. Decimal ka cube?
(1.2)³ = 1.728, decimal places teen guna.
5. Taxicab ke aur?
4104, 13832, 20683… koi bhi do alag jodi.
6. Aage kya?
Cube root: Post 7 — Cube Root Kaise Nikaalein?.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 3³ = ?
Solution dekho
27. Check ✅
Q2. Easy 10³ = ?
Solution dekho
1000. Check: 3 zero ✅
Q3. Easy Kya 81 cube?
Solution dekho
Nahi (64, 125). Check ✅
Q4. Easy (−1)³ = ?
Solution dekho
−1. Check ✅
Q5. Medium 15³ = ?
Solution dekho
225 × 15 = 3375. Check ✅
Q6. Medium (1/2)³ = ?
Solution dekho
1/8. Check ✅
Q7. Medium Rubik 4 × 4 × 4 mein kitne tukde?
Solution dekho
64. Check ✅
Q8. Medium Kaunse cubes 1000 se kam?
Solution dekho
1, 8, 27, 64, 125, 216, 343, 512, 729 (9). Check ✅
Q9. NCERT style 13832 = 18³ + ?
Solution dekho
13832 − 5832 = 8000 = 20³. Check ✅
Q10. NCERT style Cube ka aakhri digit 3, number ka?
Solution dekho
7. Check: 343 ✅
Q11. NCERT style 2-digit number ka cube max kitne digit?
Solution dekho
99³ = 970299 → 6. Check ✅
Q12. NCERT style (0.1)³ = ?
Solution dekho
0.001. Check ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Kya kisi cube ke factors hamesha odd?
Solution dekho
Nahi: 8 ke 1, 2, 4, 8 (4). Check ✅
Q2. 1729 = 1³ + 12³: 12³ − 10³ = ?
Solution dekho
1728 − 1000 = 728 = 9³ − 1³. Check ✅
Q3. Cube ka volume 27 guna: side?
Solution dekho
3 guna. Check ✅
Q4. Kaunsa number square bhi cube bhi (1 ke baad)?
Solution dekho
64 = 8² = 4³. Check ✅
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🏆 Olympiad Corner
Q1. 1 se 100 mein cubes?
Solution dekho
1, 8, 27, 64 → 4.
Q2. Agla square-cube 64 ke baad?
Solution dekho
729 = 27² = 9³.
Q3. 11³ = ?
Solution dekho
1331.
Q4. 20683 = 10³ + ?
Solution dekho
27³ (19683).
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💗 CharuMam Motivation
Ramanujan ke liye har number ek kahani tha. Aap bhi numbers se dosti karo, woh apne raaz khud batayenge! 🌟
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🟢 Master Recap
- 💡 Chaabi: Cube number = kisi number ko teen baar khud se guna: n³ = n × n × n, jo n side wale ghanaakaar (cube) ka volume hai. Square mein cheezein jodi mein aati thi, cube mein teen-teen mein: zero bhi 3, 6, 9… mein, aur prime factors bhi triplets mein.
- n³ = n × n × n, cube ka volume.
- 1–20 ke cubes table.
- Cube har digit pe khatam ho sakta.
- Zero teen-teen mein.
- Negative ka cube negative.
- 1729 = 1³ + 12³ = 9³ + 10³.
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👉 Aage Kya?
Cube ka ulta: Post 7 — Cube Root Kaise Nikaalein?.
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