✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 10 ÷ 9 ka remainder?
- 4,075 mein tens digit?
- Place value: 3,000 = 3 × ?
Answers dekho
- 1
- 7
- 1000
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 4 — Remainder Ka Khel (remainder ka khel) dekh lijiye.
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🎬 Kahani: Dukaandaar Ka Shortcut: “Digits Jodo!”
Ramesh bhaiya ki dukaan pe 5,832 toffeeyaan aayi. Unhe 9-9 ke packet banaane the. Calculator kharab! Unhone digits jode: 5 + 8 + 3 + 2 = 18. “Poore packet banenge, ek bhi nahi bachegi!” 🍭
Unka beta Aman hairaan: “Digits jodne se 9 ka kya lena-dena? Yeh to jaadu-tona lagta!”
Aaj yeh rule rataayenge nahi, samjhenge: digits ka jod 9 ke baare mein sach kyun bolta. Aur 11 ka bhi raaz kholenge. 🔓
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📖 Concept: Har Place Value = 9 Ke Dabbe + 1
💡 Is post ki ek badi baat: 10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1. Har place value 9 ke multiple se 1 zyada. Isliye har digit utne hi “bache hue” deta, aur number = (9 ka multiple) + digits ka jod. 11 ke liye: 10 = 11 − 1, 100 = 99 + 1, 1000 = 1001 − 1: baari-baari −1, +1.
Step 1: Pehle aasaan wale: 10, 5, 2 (kyun?)
Number = … + 1000d + 100c + 10b + a. Units ke alawa har place value 10 ka multiple, aur 10 = 2 × 5.
Isliye 2, 5, 10 ke liye sirf aakhri digit a dekho. Baaki sab pehle se bate hue!
WHY 9 ke liye aisa nahi? 10 khud 9 se nahi bat-ta (1 bachta). 99 aur 109 dono 9 pe khatam, par sirf 99 bat-ta.
Step 2: 9 ka rule: andar se dekho
5,832 = 5 × 1000 + 8 × 100 + 3 × 10 + 2
= 5 × (999 + 1) + 8 × (99 + 1) + 3 × (9 + 1) + 2
= (5 × 999 + 8 × 99 + 3 × 9) + (5 + 8 + 3 + 2)
Pehla hissa: 9 ka multiple. Doosra: digits ka jod = 18 = 9 × 2. Dono bate → 5,832 bata ✅
WHY kaam karta? Har digit apne place value ke 9-ke-dabbe bhar ke theek digit jitna bachaata. Sab bache hue = digits ka jod.
Step 3: Remainder bhi milta (9 se)
6,184: 6 + 1 + 8 + 4 = 19 → 1 + 9 = 10 → 1 + 0 = 1. To 6,184 ÷ 9 → 1 bachta.
Check: 9 × 687 = 6,183 → 6,184 − 6,183 = 1 ✅
3 ka rule: 9, 99, 999 sab 3 ke bhi multiple → wahi tark: digits ka jod 3 se bate to number bhi. (Remainder bhi digit-jod se.)
Step 4: 11 ka rule: Tareeka A (zyada-kam)
1 = 0 + 1 (+1), 10 = 11 − 1 (−1), 100 = 99 + 1 (+1), 1000 = 1001 − 1 (−1)…
91,828: units se: +1 wale (units, hundreds, ten-thousands): 8 + 8 + 9 = 25. −1 wale (tens, thousands): 2 + 1 = 3. Farak 25 − 3 = 22 = 11 × 2 → bata ✅
WHY? +1 wali jagah digit “zyada” deta, −1 wali “kami”. Kul zyaadti − kami = number ka 11 se bachna.
Step 5: 11 ka rule: Tareeka B (+/− signs)
Units se shuru karke baari-baari + − lagao: 5,263 → +3 − 6 + 2 − 5 = −6 → −6 + 11 = 5 bachta.
Check: 11 × 478 = 5,258 → 5,263 − 5,258 = 5 ✅
Kab kaunsa? Dono ek hi hain (B = A ko ek line mein). Bade numbers mein A (do alag jod) galti kam karta; chhote mein B tez.
🪜 Atak Gaye? Aise Socho
- 9 ya 3: digits jodo; jod bada ho to phir jodo.
- Jod 9 se bate → number 9 se. 3 se bate → number 3 se.
- 11: units se + − + − lagao.
- Jawab 0 ya 11 ka multiple → bata. Negative → 11 jodo, remainder milega.
- Hamesha ek baar asli bhaag se check karo (shuru mein).
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🔐 Deep Concept: Kyun 9 Aur 11 “Padosi” Hain 10 Ke? (Ek Hi Idea)
Hum 10 ke base mein likhte. 9 = 10 − 1 aur 11 = 10 + 1: dono 10 ke padosi!
- 10 ko 9 ki ghadi mein dekho: 10 → 1. To 100 = 10 × 10 → 1 × 1 = 1. Har power → 1. Isliye digits ka seedha jod.
- 10 ko 11 ki ghadi mein: 10 → −1. To 100 → (−1)(−1) = +1, 1000 → −1… Isliye baari-baari + −.
Yeh Post 4 — Remainder Ka Khel ka remainder wala guna niyam hai! Bas place values pe lagaaya. 🎯
Socho: agar hum base 8 mein likhte (Book 1 Ch 3 ke bases yaad?), to digit-jod ka rule 7 ke liye chalta aur baari-baari wala 9 ke liye. Rule base pe tika hai, 9 pe nahi!
Unifying idea: number = place values × digits. Divisor ki ghadi mein place values chhoti ho jaayein (1 ya −1) to rule ban jaata.
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✏️ Solved Examples
Example 1 (Easy): 73,548 kya 9 se bata? Remainder?
7 + 3 + 5 + 4 + 8 = 27 → 9 se bata → bata, remainder 0.
Check: 9 × 8,172 = 73,548 ✅
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Example 2 (Medium): 4,07,286 kya 11 se bata?
+6 − 8 + 2 − 7 + 0 − 4 = −11 → 11 ka multiple → bata.
Check: 11 × 37,026 = 4,07,286 ✅
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Example 3 (NCERT style): 2z43 (z ek digit) 9 se bate. z?
2 + z + 4 + 3 = 9 + z → 9 ya 18 chahiye → z = 0 ya 9 (do jawab!).
Check: 2043 = 9 × 227 ✅, 2943 = 9 × 327 ✅
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Ramesh bhaiya: 5,832 ki digits ka jod 18 → 648 poore packet. Aman ab kehta: “Jaadu nahi, 999 + 1 hai!” 😎
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🏠 Divisibility Rules Roz Ki Zindagi Mein
- 🛒 Packing: 9 ya 3 ke packet: digit jod se turant pata kitne bachenge.
- 🧾 Bill check: purane munim “nau ka niyam” (digit jod) se jod-guna ki galti pakadte the.
- 🎟️ Group mein baantna: 3 teams barabar? Bachchon ki ginti ka digit jod 3 se bate?
- 📚 ISBN/barcode: aakhri digit ek “check digit” hai, jo 10/11 ke remainder se banta.
- 🎲 Number magic: koi number lo, digits ulta karo, ghatao: jawab hamesha 9 ka multiple!
Magic try karo: 742 − 247 = 495 → 4 + 9 + 5 = 18 ✅ Kyun? Dono numbers ka digit-jod same, to farak 9 ka multiple.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| 9 / 3 | ✅ Digits ka jod | Remainder bhi |
| 11 | ✅ Units se + − + − | Ya zyada − kam |
| 2, 5, 10 | ✅ Aakhri digit | 10 = 2 × 5 |
| Missing digit | ✅ Jod se equation | Do jawab ho sakte |
| Remainder 11 se | ✅ Negative → +11 | Phir chhota karo |
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❌→✅ Worked Mistake
Sawaal: 5,263 ko 11 se bhaag pe kitna bachta?
❌ Galat: bayein (left) se + lagaaya: +5 − 2 + 6 − 3 = 6. Galat remainder!
✅ Sahi: units digit ki place value 1 (+1), isliye units se +: +3 − 6 + 2 − 5 = −6 → −6 + 11 = 5. Check: 11 × 478 = 5,258, bacha 5 ✅ (Bayein se shuru karne pe 4-digit number mein saare sign ulte ho jaate: “bata/nahi” to sahi aata, remainder galat.)
Doosri common galti: 9 ka rule 11 pe lagaana (digits seedha jodna). 11 ki ghadi mein 10 = −1 hai, +1 nahi.
🧠 Rule of thumb: 9 ke liye seedha jod; 11 ke liye units se shuru + − + −.
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📝 10 MCQs
Q1. 4,536 kya 9 se bata?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 4 + 5 + 3 + 6 = 18, 9 se bata ✅
- (b) galat: Jod 18, 9 ka multiple hai.
- (c) galat: 9 se bata to 3 se bhi bata.
- (d) galat: Digit jod se pakka pata chalta.
Q2. 2,350 ka 9 se remainder?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 2 + 3 + 5 + 0 = 10 → 1 ✅
- (b) galat: 2 ka hisaab galat nikala.
- (c) galat: 10 ko phir jodo: 1 + 0 = 1.
- (d) galat: Jod 10, 9 ka multiple nahi.
Q3. 100 ko 11 se bhaag pe?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 100 = 99 + 1 → 1 bacha ✅
- (b) galat: 10 tens wali jagah ka hai.
- (c) galat: −1 to 10 aur 1000 ke liye.
- (d) galat: 99 = 11 × 9, 1 bachta hai.
Q4. 3,861 kya 11 se?
Solution dekho
Sahi jawab: (a)
- (a) sahi: +1 − 6 + 8 − 3 = 0 → bata ✅
- (b) galat: 0 aaya, matlab 11 se bata.
- (c) galat: 3 se bhi bata, par 11 se bhi.
- (d) galat: Dekho, 0 aaya to bata hai.
Q5. 9 ka rule kyun chalta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Har place value 9 ke multiple + 1 ✅
- (b) galat: Size ka koi lena-dena nahi.
- (c) galat: 11 bhi odd par rule alag.
- (d) galat: Har niyam ka ek kyun hota hai.
Q6. 48a2 3 se bate. a = 0 se 9 mein kitne?
Solution dekho
Sahi jawab: (b)
- (a) galat: Do se zyada values milti hain.
- (b) sahi: 14 + a: a = 1, 4, 7 → teen ✅
- (c) galat: a = 10 digit nahi, to 4 nahi.
- (d) galat: Sirf ek nahi, teen hain.
Q7. 6,72,9b 9 se bate. b?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 6 + 7 + 2 + 9 + b = 24 + b → 3 ✅
- (b) galat: 24 + 6 = 30, 9 se nahi.
- (c) galat: 24, 9 ka multiple nahi.
- (d) galat: 24 + 9 = 33, 9 se nahi.
Q8. Kaunsa 11 se nahi bata?
Solution dekho
Sahi jawab: (c)
- (a) galat: 121 = 11 × 11, bata hai.
- (b) galat: 1331 = 11 × 121, bata hai.
- (c) sahi: +6 − 3 + 0 − 2 = 1 → nahi bata ✅
- (d) galat: +2 − 6 + 8 − 4 = 0, bata hai.
Q9. Base 8 mein digit-jod rule kiske liye?
Solution dekho
Sahi jawab: (b)
- (a) galat: 9 base 10 ke liye tha.
- (b) sahi: 7 = 8 − 1, padosi number ✅
- (c) galat: 8 ke liye aakhri digit dekho.
- (d) galat: 10 base 8 mein khaas nahi.
Q10. 742 − 247 ka digit jod?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 495 → 4 + 9 + 5 = 18 ✅
- (b) galat: 11 ka rule yahan nahi lagta.
- (c) galat: 495 ka jod 18, 9 nahi.
- (d) galat: 27 to bahut zyada hai.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. 3 ka rule 9 jaisa kyun?
9, 99, 999 sab 3 se bhi bat-te. To wahi tark: number = 3 ka multiple + digits ka jod.
2. Digit jod ko baar-baar jodna sahi?
Haan: jod bhi ek number hai, uska bhi wahi niyam. Aakhir mein ek digit = remainder (9 aaye to 0).
3. 11 ke rule mein units se hi kyun?
Kyunki units ki place value 1 (= +1). Wahan se + shuru karna remainder sahi deta.
4. Kya ulta bhi sach: digit jod 9 se bate → number?
Haan, dono taraf! Number = 9 ka multiple + jod, to dono saath bat-te ya saath nahi.
5. 7 ka koi rule?
Hai, par lamba (1001 = 7 × 11 × 13 se). Aage seekhoge.
6. Yeh “why” kyun zaroori?
Rule bhool sakte ho, “why” nahi. Aur “why” se naye rules khud bana sakte ho (jaise base 8 wala).
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 837 kya 9 se?
Solution dekho
18 → haan. Check: 9 × 93 = 837 ✅
Q2. Easy 1,234 ka 9 se remainder?
Solution dekho
10 → 1. Check: 9 × 137 = 1,233 ✅
Q3. Easy 5,016 kya 3 se?
Solution dekho
12 → haan. Check: 3 × 1,672 ✅
Q4. Easy 616 kya 11 se?
Solution dekho
+6 − 1 + 6 = 11 → haan. Check: 11 × 56 ✅
Q5. Medium 25,817 ka 11 se remainder?
Solution dekho
+7 − 1 + 8 − 5 + 2 = 11 → 0. Check: 11 × 2,347 = 25,817 ✅
Q6. Medium 6,403 ka 11 se remainder?
Solution dekho
+3 − 0 + 4 − 6 = 1 → 1. Check: 11 × 582 = 6,402 ✅
Q7. Medium Sabse chhota 4-digit number jo 9 se bate?
Solution dekho
1,008 (jod 9). Check: 9 × 112 ✅
Q8. Medium 7a5 3 se bate: a ki values?
Solution dekho
12 + a → 0, 3, 6, 9. Check: 735 = 3 × 245 ✅
Q9. NCERT style 9 se bata number, digits ulta: phir bhi 9 se?
Solution dekho
Hamesha: digit jod same. Check: 783 → 387 = 9 × 43 ✅
Q10. NCERT style 6-digit number abcabc 11 se?
Solution dekho
Hamesha: abc × 1001 = abc × 7 × 11 × 13. Check: 123123 = 11 × 11193 ✅
Q11. NCERT style 2-digit number aur uska ulta: jod kis ka multiple?
Solution dekho
(10a + b) + (10b + a) = 11(a + b) → 11. Check: 47 + 74 = 121 ✅
Q12. NCERT style 5,000 ke sabse paas 9 ka multiple?
Solution dekho
5 → 5 bacha → 4,995 (5 door) ya 5,004 (4 door) → 5,004. Check: 9 × 556 ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Ek 3-digit number, uske digits ka ulta, farak hamesha kis ka multiple?
Solution dekho
99(a − c) → 99 (9 aur 11). Check: 853 − 358 = 495 = 99 × 5 ✅
Q2. Kya 9 aur 11 dono ke rule se 99 ka rule ban sakta?
Solution dekho
Haan: 9 aur 11 co-prime → dono se bate to 99 se. Ya do-do digits ka jod (100 = 99 + 1). Check: 4,356 → 43 + 56 = 99 ✅
Q3. 3,000 se 3,100 ke beech 9 ke kitne multiples?
Solution dekho
3,006 se 3,096 → 11. Check: (3096 − 3006)/9 + 1 = 11 ✅
Q4. Sabse chhota 9 ka multiple jiski saari digits even?
Solution dekho
Jod 18 chahiye, even digits: 288. Check: 9 × 32 ✅
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🏆 Olympiad Corner
Q1. 1 se 99 tak saare numbers likh do (123456789101112…). 9 se remainder?
Solution dekho
Digit jod 900 → 0.
Q2. 10¹⁰ − 1 ka digit jod?
Solution dekho
Das 9 → 90.
Q3. 7-digit sab 1 wala number 1111111 ko 11 se?
Solution dekho
+1 −1… → 1.
Q4. Har 3-digit 99 ke multiple (198, 297…) ka beech wala digit?
Solution dekho
Hamesha 9: 11 ka rule a − m + c = 0 aur 9 ka rule saath.
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💗 CharuMam Motivation
Aman ka “jaadu-tona” ab ek saaf saboot hai. Jab aap rule ka “kyun” samajhte ho, aap usse naye rules bana sakte ho. Yahi asli maths hai! 🧠
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🟢 Master Recap
- 💡 Chaabi: 10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1. Har place value 9 ke multiple se 1 zyada. Isliye har digit utne hi “bache hue” deta, aur number = (9 ka multiple) + digits ka jod. 11 ke liye: 10 = 11 − 1, 100 = 99 + 1, 1000 = 1001 − 1: baari-baari −1, +1.
- 10, 100, 1000 = 9 ke multiple + 1.
- Number = 9 ka multiple + digits ka jod.
- 9 aur 3: digit jod; remainder bhi.
- 11: place values +1, −1, +1…
- Units se + − + −; negative → +11.
- Rule base pe tika: base 8 mein 7 aur 9.
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👉 Aage Kya?
4, 8, 6, 24 ke rules aur digital root ka kamaal: agla post Post 6 — Divisibility: 4, 8 Aur Digital Root.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬