✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 13² = ?
- 7 × 7 ka aakhri digit?
- Kya 30 even hai ya odd?
Answers dekho
- 169
- 9 (49)
- Even
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, pichla post Post 1 — Square Numbers Kya Hain? dekh lijiye.
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🎬 Kahani: Riya Ka 2 Second Wala Jawab
Maths quiz mein sawaal aaya: “Kaunsa perfect square NAHI hai: 2032, 2048, 1027, 1024?” Sab bachche calculator dhoondhne lage. Riya ne 2 second mein bola: “Pehle teeno! Sirf 1024 ho sakta hai.” 😲
Na guna kiya, na root nikaala. Bas aakhri digit dekha. Kaunsa jaadu? Chaliye seekhte hain.
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📖 Concept: Aakhri Digit Ka Niyam Aur Uske Saathi
💡 Is post ki ek badi baat: Square ka aakhri digit sirf number ke aakhri digit pe tika hai, isliye square kabhi 2, 3, 7, 8 pe khatam nahi hota. Yeh niyam “square nahi hai” turant bata deta hai, par “square hai” sabit nahi karta.
Step 1: Aakhri digit ka table
| Number ka aakhri digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Square ka aakhri digit | 0 | 1 | 4 | 9 | 6 | 5 | 6 | 9 | 4 | 1 |
WHY? 23 × 23 mein aakhri digit sirf 3 × 3 = 9 se aata hai; baaki sab tens aur upar jaata hai. Isliye sirf 0–9 ke squares dekhne kaafi.
Step 2: Sirf 0, 1, 4, 5, 6, 9
Table mein 2, 3, 7, 8 kabhi nahi aate. To jo number 2, 3, 7, 8 pe khatam ho, woh pakka square nahi.
Par dhyaan: 26 aakhri digit 6, phir bhi square nahi. Niyam sirf “nahi” ke liye pakka hai.
Step 3: Zero hamesha jodi mein
10² = 100, 20² = 400, 100² = 10000. Zero 2, 4, 6… (even) aate hain.
WHY? Number ke aakhri k zero hon, to square mein k + k = 2k zero. Isliye 1000, 250, 40 jaisa koi square nahi (odd zero).
Step 4: Even-odd
Even ka square even (4² = 16), odd ka square odd (7² = 49). WHY? Even × even = even, odd × odd = odd.
Step 5: Do squares ke beech kitne numbers
4² = 16 aur 5² = 25 ke beech: 17 se 24 → 8 = 2 × 4 numbers. n² aur (n + 1)² ke beech 2n non-square numbers.
WHY? (n + 1)² − n² = 2n + 1 (Post 3 mein picture). Beech mein usse 1 kam = 2n.
🪜 Atak Gaye? Aise Socho
- Sabse pehle aakhri digit dekho: 2, 3, 7, 8? Seedha “nahi”.
- Aakhri mein zero? Gino: odd zero → nahi.
- Pass ho gaya? Ab do nazdeeki squares ke beech dekho.
- Phir bhi shak? Post 4 ka prime factorisation.
- Table yaad karo: 1↔9 → 1, 2↔8 → 4, 3↔7 → 9, 4↔6 → 6.
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🔐 Deep Concept: 5 Pe Khatam Hone Wale Squares Ka Shortcut
35² = ? Pehla digit 3, uske agle se guna: 3 × 4 = 12, peeche 25 → 1225. 65² = 6 × 7 = 42 → 4225.
WHY yeh kaam karta? (10a + 5)² = 100a² + 100a + 25 = 100 × a(a + 1) + 25. Isliye aakhri do digit hamesha 25, aur aage a(a + 1).
Digit ginti: 1-digit squares 3 (1, 4, 9), 2-digit 6 (16 se 81), 3-digit 22 (100 se 961). Jaise-jaise number bade, beech ke gap (2n + 1) bhi bade, isliye squares “dur-dur” hote jaate.
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✏️ Solved Examples
Example 1 (Easy): Kaunse square nahi: 2032, 2048, 1027?
Aakhri digit 2, 8, 7: teeno square nahi.
Check: 45² = 2025, 46² = 2116; 2032, 2048 beech mein ✅
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Example 2 (Medium): Kya 4000 perfect square?
3 zero (odd) → nahi.
Check: 63² = 3969, 64² = 4096; 4000 beech mein ✅
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Example 3 (NCERT style): NCERT style: 12² aur 13² ke beech kitne non-square numbers?
2n = 2 × 12 = 24.
Check: 144 se 169 tak: 168 − 145 + 1 = 24 ✅
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Riya ka raaz: “Aakhri digit pehle dekho. 2, 3, 7, 8 = seedha bahar!” ⚡
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🏠 Yeh Niyam Kahan Kaam Aata
- 📝 MCQ exam: galat options turant hatao.
- 🧮 Calculator check: 57² ka jawab 3248 aaya? 7 ka square 9 pe khatam, to galat (sahi 3249).
- 🔐 Password paheli: “4-digit square jo 7 pe khatam” = koi nahi.
- 🏠 Tiles: 1000 tiles se square farsh nahi banega (odd zero).
- 🎯 Mental maths: 85² = 8 × 9 | 25 = 7225 seconds mein.
Pehle sasta test (digit), phir mehenga (calculation).
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Aakhri digit 2, 3, 7, 8 | ✅ Seedha “square nahi” | Table mein yeh digit nahi |
| Aakhri digit 0, 1, 4, 5, 6, 9 | ⚠️ Shayad, aur check karo | 26 bhi 6 pe khatam |
| Aakhri mein odd zero | ✅ “Square nahi” | Zero jodi mein aate |
| Number 5 pe khatam ka square | ✅ a(a + 1) | 25 | 100a(a + 1) + 25 |
| Digit niyam se “square hai” kehna | ❌ Kabhi nahi | Niyam sirf rad karta |
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❌→✅ Worked Mistake
Sawaal: Kya 156 perfect square hai?
❌ Galat: “6 pe khatam, to square hai.”
✅ Sahi: 12² = 144, 13² = 169; 156 beech mein → nahi. Check: 156 = 2 × 2 × 3 × 13, 3 aur 13 akele ✅
🧠 Rule of thumb: Aakhri digit ka niyam darwaza band karta hai, kholta nahi.
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📝 10 MCQs
Q1. Square kis digit pe khatam nahi ho sakta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 7 table mein kabhi nahi aata ✅
- (b) galat: 6 aata hai, jaise 16 aur 36.
- (c) galat: 5 aata hai, jaise 25, isliye yeh galat.
- (d) galat: 9 aata hai, jaise 49, isliye yeh galat.
Q2. 47² ka aakhri digit?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 7 × 7 = 49, aakhri digit 9 ✅
- (b) galat: 4 to 2 ya 8 ke square ka hai.
- (c) galat: 7 kisi square ka aakhri digit nahi.
- (d) galat: 1 to 1 ya 9 ke square ka hai.
Q3. Kaunsa pakka square nahi?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 8 pe khatam, kabhi square nahi ✅
- (b) galat: 1024 = 32², square hai.
- (c) galat: 196 = 14², square hai.
- (d) galat: 441 = 21², square hai.
Q4. Kya 90000 square ho sakta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 300² = 90000, 4 zero even ✅
- (b) galat: Zero 4 hain, even, odd nahi.
- (c) galat: 9 se to squares khoob shuru hote.
- (d) galat: Saaf niyam hai, isliye pata hai.
Q5. 7² aur 8² ke beech non-squares?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: 2n = 14 (50 se 63) ✅
- (b) galat: 15 tab hota jab ek square bhi gina.
- (c) galat: 7 sirf n hai, 2n chahiye.
- (d) galat: 16 bahut zyada hai, galat.
Q6. 45² = ?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 4 × 5 = 20, peeche 25 → 2025 ✅
- (b) galat: 16 to 4 × 4 se aata, galat shortcut.
- (c) galat: Square 5 ka aakhri do 25 hi hote.
- (d) galat: 21 ka koi aadhar nahi, galat.
Q7. Odd number ka square?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi jawab: Odd × odd = odd ✅
- (b) galat: Even sirf even ke square mein.
- (c) galat: Niyam pakka hai, kabhi bhi nahi.
- (d) galat: Zero sirf 0 ka square hai.
Q8. 3-digit perfect squares kitne?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 10² se 31² tak, 22 squares ✅
- (b) galat: 21 tab jab ek chhoot gaya, galat.
- (c) galat: 10 bahut kam hai, galat.
- (d) galat: 90 to 2-digit numbers ki ginti hai.
Q9. Calculator ne 63² = 3968 dikhaya. Sahi?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 3 × 3 = 9, sahi 3969 ✅
- (b) galat: 8 pe square khatam nahi hota.
- (c) galat: 4 to 2 ya 8 ka square deta.
- (d) galat: Digit niyam se turant pata chalta.
Q10. 6 pe khatam hone wala number?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 36 bhi, 26 bhi; aur check chahiye ✅
- (b) galat: 26 pe khatam bhi square nahi.
- (c) galat: 16 aur 36 to square hain.
- (d) galat: Even hona square ki guarantee nahi.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Square 2, 3, 7, 8 pe kyun nahi?
0–9 ke squares ke aakhri digit 0, 1, 4, 5, 6, 9 hi aate.
2. Kya 100 ka square 4 zero?
Haan, 10000. Zero do guna.
3. 2n wala niyam yaad kaise?
Beech mein 2n, gap 2n + 1. Post 3 ki odd numbers wali picture.
4. Kya decimal ka bhi square?
Haan: (1.5)² = 2.25. Decimal places do guna.
5. Niyam pass, to?
Root ka andaaza ya prime factorisation (Post 4 — Square Root Kya Hai?).
6. Aage kya?
Odd numbers se squares: Post 3 — Odd Numbers Aur Squares Ka Rishta.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy Kaunsa square nahi: 343, 361, 400?
Solution dekho
343 (3 pe). Check: 19² = 361, 20² = 400 ✅
Q2. Easy 58² ka aakhri digit?
Solution dekho
4. Check: 8 × 8 = 64 ✅
Q3. Easy Kya 1000 square?
Solution dekho
Nahi, 3 zero. Check ✅
Q4. Easy 25² = ?
Solution dekho
625. Check: 2 × 3 = 6 | 25 ✅
Q5. Medium 10² aur 11² ke beech non-squares?
Solution dekho
20. Check: 101 se 120 ✅
Q6. Medium 95² = ?
Solution dekho
9 × 10 = 90 → 9025. Check ✅
Q7. Medium Kya 1027 square?
Solution dekho
Nahi, 7 pe. Check: 32² = 1024 ✅
Q8. Medium 2-digit squares likho.
Solution dekho
16, 25, 36, 49, 64, 81. Check: 6 ✅
Q9. NCERT style Kya 222222 square?
Solution dekho
Nahi, 2 pe. Check ✅
Q10. NCERT style Kya 250 square? Do kaaran.
Solution dekho
Nahi: ek zero (odd); 15² = 225, 16² = 256. Check ✅
Q11. NCERT style Kaunse digit pe khatam squares sabse zyada (1–100)?
Solution dekho
1, 4, 9, 16, 25, 36, 49, 64, 81, 100: 1, 6, 9, 4 do-do baar. Check ✅
Q12. NCERT style 4-digit squares kitne?
Solution dekho
32² se 99²: 68. Check: 99 − 32 + 1 ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Square ka aakhri digit 6, to number ka aakhri?
Solution dekho
4 ya 6. Check: 14² = 196, 16² = 256 ✅
Q2. Square ke aakhri do digit 25, number?
Solution dekho
5 pe khatam. Check: 35² = 1225 ✅
Q3. Kya do lagataar numbers dono square?
Solution dekho
Sirf 0 aur 1. Gap 2n + 1 ≥ 3. Check ✅
Q4. 1 se 1000 tak 9 pe khatam squares?
Solution dekho
3 ya 7 pe khatam roots ≤ 31: 3, 7, 13, 17, 23, 27 → 6. Check ✅
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🏆 Olympiad Corner
Q1. Kya 10ⁿ + 2 square ho sakta?
Solution dekho
Nahi, 2 pe khatam.
Q2. Aakhri digit 5 wale square ka tens digit?
Solution dekho
Hamesha 2.
Q3. 9999² ka aakhri digit?
Solution dekho
1.
Q4. 11² se 20² mein kitne 6 pe khatam?
Solution dekho
14, 16 → 2.
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💗 CharuMam Motivation
Riya ne calculator ke bina jeet li, sirf ek pattern se. Pattern dekhne ki aadat daaliye, maths tez aur mazedaar ho jayegi! 🌟
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🟢 Master Recap
- 💡 Chaabi: Square ka aakhri digit sirf number ke aakhri digit pe tika hai, isliye square kabhi 2, 3, 7, 8 pe khatam nahi hota. Yeh niyam “square nahi hai” turant bata deta hai, par “square hai” sabit nahi karta.
- Square ka aakhri digit: 0, 1, 4, 5, 6, 9.
- 2, 3, 7, 8 pe khatam = square nahi.
- Zero hamesha even ginti mein.
- Even² even, odd² odd.
- n² aur (n + 1)² ke beech 2n numbers.
- 5 wala shortcut: a(a + 1) | 25.
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👉 Aage Kya?
Ab odd numbers aur triangular numbers ka jaadu: Post 3 — Odd Numbers Aur Squares Ka Rishta.
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