✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 38 ÷ 5: quotient aur remainder?
- Remainder divisor se bada ho sakta?
- 5k ka matlab?
Answers dekho
- 7 aur 3
- Nahi, hamesha chhota
- 5 ke saare multiples
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 3 — Hamesha, Kabhi-Kabhi Ya Kabhi Nahi? dekh lijiye.
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🎬 Kahani: Halwai Chacha Ke Laddoo
Halwai chacha laddoo 7-7 ke dabbon mein bharte. Har baar 4 laddoo bach jaate. Aaj unke paas do tokriyaan hain: dono mein laddoo ginne ka time nahi, par dono mein “7 ke dabbe, 4 bache”. 🍬
Chhotu poochta: “Dono tokri milaa doon to kitne bachenge?” Chacha: “8!” Chhotu hansa: “8 bache to ek aur dabba ban jaayega na?”
Bina gine, bachne wale laddoo ka hisaab kaise? Aaj remainder ka khel! 🎯
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📖 Concept: Number = Poore Dabbe + Bache Hue
💡 Is post ki ek badi baat: Har number ko divisor × (poore dabbe) + remainder likh sakte: jaise 7k + 4. Jodne-ghataane-guna mein sirf remainders ka hisaab karo, aur aakhir mein divisor se dobara dabbe banao.
Step 1: Remainder wale numbers ka roop
7 se bhaag pe 4 bache: 4, 11, 18, 25, 32… = 7k + 4 (k = 0, 1, 2…).
WHY? Har number = kuch poore 7 ke dabbe (7k) + 4 bache. k badhao, ek dabba aur.
Wahi numbers 7k − 3 bhi (k = 1, 2…): 4 = 7 − 3, 11 = 14 − 3. “4 zyada” = “3 kam” agle dabbe se!
Step 2: Remainders jodo
(7a + 4) + (7b + 4) = 7(a + b) + 8 = 7(a + b + 1) + 1.
Chhotu sahi: 8 bache matlab ek aur dabba, 1 bacha.
WHY kaam karta? Poore dabbe poore hi rehte; sirf bache hue milte. Unme se phir dabba banao.
Step 3: Ghataav aur guna
543 ÷ 8 → 7 bache; 230 ÷ 8 → 6 bache.
- Jod: 7 + 6 = 13 → 13 − 8 = 5
- Farak: 7 − 6 = 1
- Guna: 7 × 6 = 42 → 42 ÷ 8 → 2
Check: 773 = 8 × 96 + 5 ✅, 313 = 8 × 39 + 1 ✅, 543 × 230 = 124890 = 8 × 15611 + 2 ✅
Farak negative aaye (2 − 5 = −3) to divisor jodo: −3 + 8 = 5.
Step 4: Do shart: 3 se 2 bache, 5 se bhi 2 bache
Dono ka matlab: number − 2, 3 aur 5 dono se bate → LCM 15 se → 15k + 2: 2, 17, 32, 47…
Aur 4 se bhi poora bate? 32 ✅ (sabse chhota, 0 se bada).
Step 5: “Ek kam” wali trick
5 se 4 bache, 6 se 5 bache, 7 se 6 bache. Kya pattern? Har baar ek kam poora dabba!
To number + 1 → 5, 6, 7 sab se bate → LCM 210 → number = 209.
Kab kaunsa? Remainder sab same: “number − r” ka LCM. Remainder har baar divisor se ek kam: “number + 1” ka LCM. Alag-alag: list banao (15k + 2 jaisa) aur check karo.
🪜 Atak Gaye? Aise Socho
- Remainder r, divisor d → number = dk + r.
- Jodna/ghataana: sirf r jodo/ghatao, phir d se dabba banao.
- Negative aaye to d jodo.
- Do shart, same remainder: number − r ko LCM se.
- Remainder = d − 1 har baar: number + 1 ko LCM se.
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🔐 Deep Concept: Ghadi Ka Hisaab: Remainder Ek Gol Duniya Hai
Ghadi mein 9 baje + 5 ghante = 2 baje, 14 nahi! Ghadi 12 pe wapas ghoomti. Yahi remainder: 14 ÷ 12 → 2. ⏰
Remainder ki duniya gol hai: 7 se bhaag mein sirf 0, 1, 2, 3, 4, 5, 6. 7 aate hi phir 0. Isliye 4 + 4 = 8 → 1.
Kyun bade numbers chhod sakte? Kyunki poore dabbe (7k) har jod, ghataav, guna mein poore hi rehte: 7a × 7b, 7a × 4, sab mein 7 factor. Sirf “bache × bache” naya hissa.
Unifying idea: Post 2 — Even-Odd Ka Khel: Parity ki parity bhi yahi tha (2 wali ghadi: sirf 0 aur 1). Aage Post 5 — Divisibility Rule: 9, 3 Aur 11 ke divisibility rules bhi 9 aur 11 ki ghadi hain! 🎯
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✏️ Solved Examples
Example 1 (Easy): 6 se bhaag pe 5 bache: aisa number algebra mein? Aur doosra roop?
6k + 5 ya 6k − 1.
Check: k = 2 → 17; 17 = 6 × 2 + 5 ✅ aur 18 − 1 = 17 ✅
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Example 2 (Medium): Chaar numbers, har ek 4 se bhaag pe 3 chhodta. Jod 4 ka multiple?
(4a + 3) + (4b + 3) + (4c + 3) + (4d + 3) = 4(a + b + c + d) + 12 = 4(… + 3) → hamesha.
Check: 3 + 7 + 11 + 15 = 36 = 4 × 9 ✅
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Example 3 (NCERT style): Sabse chhota number: 3 se 2, 4 se 3, 5 se 4 bache?
Har baar ek kam → number + 1 = LCM(3, 4, 5) = 60 → 59.
Check: 59 = 3 × 19 + 2 = 4 × 14 + 3 = 5 × 11 + 4 ✅. Kyun sabse chhota? number + 1 ko 60 ka multiple hona hi tha, aur 60 sabse chhota.
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Chacha ke laddoo: 4 + 4 = 8 → ek naya dabba, 1 bachega. Chacha ne Chhotu ko inaam mein woh ek laddoo de diya! 😋
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🏠 Remainder Roz Ki Zindagi Mein
- ⏰ Ghadi: 30 ghante baad kitne baje? 30 ÷ 12 → 6 bache, 6 ghante aage.
- 📅 Din: aaj Sunday, 100 din baad? 100 ÷ 7 → 2 bache → Tuesday.
- 🚌 Bus: 50 bachche, 12 seat ki van: 4 van, 2 bache → ek aur van chahiye.
- 🍕 Pizza ke tukde: 26 tukde, 8 dost: 3-3 mile, 2 bache.
- 💳 Card number: bank bahut se numbers ka remainder dekh ke galat number pakadte (check digit).
“100 din baad kaunsa din?” jaise sawaal dost se poochho: remainder se turant jawab!
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Remainder r, divisor d | ✅ dk + r | Ya dk − (d − r) |
| Jod/ghataav | ✅ Sirf remainders | Phir dabba banao |
| Same remainder, do divisor | ✅ LCM·k + r | Number − r |
| Har baar d − 1 bacha | ✅ Number + 1 ko LCM | Ek kam trick |
| Alag-alag remainders | ✅ List + check | Chhota se shuru |
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❌→✅ Worked Mistake
Sawaal: 9 se bhaag pe 7 aur 8 bachne wale do numbers ka jod 9 se bhaag pe?
❌ Galat: “7 + 8 = 15 bachega.” Remainder 9 se bada nahi ho sakta!
✅ Sahi: 15 = 9 + 6 → 6. Check: 16 + 17 = 33 = 9 × 3 + 6 ✅
Doosri common galti: remainder 3 (÷5) wale numbers ko 3k + 5 likhna. Ulta! Divisor ke saath k: 5k + 3.
🧠 Rule of thumb: Remainder hamesha divisor se chhota: bada aaye to ek dabba aur banao.
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📝 10 MCQs
Q1. 8 se 5 bachne wale numbers?
Solution dekho
Sahi jawab: (b)
- (a) galat: Ulta: divisor 8 ke saath k chahiye.
- (b) sahi: Poore 8 ke dabbe + 5 ✅
- (c) galat: 8k − 5 se 3 bachta hai.
- (d) galat: 5k to 5 ke multiples hain.
Q2. 11k − 2 se bhaag (11) pe kitna bachta?
Solution dekho
Sahi jawab: (b)
- (a) galat: −2 ka matlab 9 bachta.
- (b) sahi: 11 − 2 = 9, itna bachta ✅
- (c) galat: Remainder 11 nahi ho sakta.
- (d) galat: Zero tab jab sirf 11k ho.
Q3. 5 se 3 aur 4 bache: jod 5 se?
Solution dekho
Sahi jawab: (b)
- (a) galat: 7 to 5 se bada hai, dabba banao.
- (b) sahi: 7 − 5 = 2, yahi bacha ✅
- (c) galat: 3 + 4 = 7, 1 nahi bachta.
- (d) galat: Zero tab jab jod 5 ka multiple.
Q4. 9 se 2 aur 6 bache: farak (pehla − doosra)?
Solution dekho
Sahi jawab: (b)
- (a) galat: Negative mein 9 jodna padta.
- (b) sahi: 2 − 6 = −4, phir +9 = 5 ✅
- (c) galat: 4 to 6 − 2 hai, ulta ghataaya.
- (d) galat: 3 ka koi hisaab nahi bana.
Q5. 7 se 3 aur 5 bache: guna 7 se?
Solution dekho
Sahi jawab: (b)
- (a) galat: 15 to 7 se bada, dabba banao.
- (b) sahi: 3 × 5 = 15 → 14 hata ke 1 ✅
- (c) galat: 8 to jod hai, guna nahi.
- (d) galat: 15 7 ka multiple nahi.
Q6. 3 se 1 aur 4 se 1 bache: roop?
Solution dekho
Sahi jawab: (a)
- (a) sahi: LCM 12 ke multiples, phir +1 ✅
- (b) galat: 7 = jod hai, LCM nahi.
- (c) galat: 12k mein kuch nahi bachta.
- (d) galat: Yeh dono shart poori nahi karta.
Q7. Aaj Wednesday, 50 din baad?
Solution dekho
Sahi jawab: (b)
- (a) galat: 50 ÷ 7 → 1 bacha, ek din aage.
- (b) sahi: 1 din aage → Thursday ✅
- (c) galat: Friday to 2 din aage hota.
- (d) galat: Monday to peeche ki taraf hai.
Q8. 5 se 4, 6 se 5 bache, sabse chhota?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Number + 1 = LCM 30 → 29 ✅
- (b) galat: 30 to poora bat-ta hai.
- (c) galat: 31 se 5 ke bhaag pe 1 bachta.
- (d) galat: 59 bhi chalega par bada hai.
Q9. 6k + 4 kya hamesha even?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 2(3k + 2): hamesha even ✅
- (b) galat: 2 bahar nikal aata hai.
- (c) galat: k kuch bhi ho, even hi aata.
- (d) galat: Algebra se saaf pata chalta.
Q10. 12 ghante ki ghadi, 8 baje + 27 ghante?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 8 + 27 = 35 → 35 − 24 = 11 ✅
- (b) galat: 3 ka hisaab galat nikla.
- (c) galat: Ghadi mein 35 baje nahi hote.
- (d) galat: 27 ghante poora chakkar nahi.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Remainder ka niyam kyun chalta?
Divisor ke poore dabbe har operation mein divisor ke multiple hi rehte. Bas bache hue naya hissa banaate.
2. 7k + 4 aur 7k − 3 same kaise?
Dono 4, 11, 18… dete: bas k ki shuruaat alag (0 se ya 1 se).
3. Negative remainder?
Aam taur pe remainder 0 se divisor − 1 tak. −3 aaye to divisor jodo.
4. Ek kam trick har baar?
Jab har remainder = divisor − 1. Warna list waala tareeka.
5. Bina LCM ke?
Sabse bade divisor wali list banao (15k + 2…) aur baaki shart check karo.
6. Aage kya?
Class 9-10 mein Euclid ka division lemma: a = bq + r. Yahi!
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 4 se 1 bachne wale teen numbers aur roop?
Solution dekho
1, 5, 9 → 4k + 1. Check: 9 = 4 × 2 + 1 ✅
Q2. Easy 10k + 7 ko 10 se bhaag pe?
Solution dekho
7. Check: 37 ✅
Q3. Easy 6 se 2 aur 3 bache: jod 6 se?
Solution dekho
5. Check: 8 + 9 = 17 = 12 + 5 ✅
Q4. Easy Aaj Friday, 20 din baad?
Solution dekho
20 ÷ 7 → 6 → Thursday. Check ✅
Q5. Medium 9 se 5 aur 7 bache: jod aur guna 9 se?
Solution dekho
Jod 12 → 3; guna 35 → 8. Check: 14 + 16 = 30 → 3 ✅
Q6. Medium 2 se 1, 3 se 1, 4 se 1 bache, 5 se poora: sabse chhota?
Solution dekho
12k + 1: 13, 25 → 25. Check ✅
Q7. Medium Teen numbers, sab 5 se 3 chhodte. Jod 5 se?
Solution dekho
9 → 4. Check: 3 + 8 + 13 = 24 → 4 ✅
Q8. Medium 4 se 3, 6 se 5 bache, sabse chhota?
Solution dekho
Number + 1 = 12 → 11. Check ✅
Q9. NCERT style 2 se 1, 3 se 2, 4 se 3, 5 se 4, 6 se 5: sabse chhota?
Solution dekho
LCM 60 → 59. Check: 59 = 6 × 9 + 5 ✅
Q10. NCERT style 100 se kam number: 3 se 2, 5 se 2, 7 se poora?
Solution dekho
15k + 2: 2, 17, 32, 47, 62, 77 → 77. Check: 77 = 7 × 11 ✅
Q11. NCERT style Kaunse numbers 4 se 2 aur 3 se 2 chhodte? Roop?
Solution dekho
12k + 2. Check: 14 = 4 × 3 + 2 = 3 × 4 + 2 ✅
Q12. NCERT style 8 se 7 bachne wale ka square 8 se?
Solution dekho
7 × 7 = 49 → 1. Check: 15² = 225 = 8 × 28 + 1 ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Har odd number ka square 8 se bhaag pe kya chhodta?
Solution dekho
(2n + 1)² = 4n(n + 1) + 1, n(n + 1) even → 1. Check: 9² = 81 = 80 + 1 ✅
Q2. 2¹⁰⁰ ko 3 se bhaag pe?
Solution dekho
2 → 2, 4 → 1, 8 → 2… even power → 1. Check: 2⁴ = 16 → 1 ✅
Q3. Ek number ko 12 se bhaag pe 7 bacha. Use 4 se bhaag pe?
Solution dekho
12k + 7 = 4(3k + 1) + 3 → 3. Check: 19 → 3 ✅
Q4. Kya koi number 6 se 2 aur 9 se 4 chhod sakta?
Solution dekho
6k + 2 ko 3 se bhaag pe 2 bachta; 9m + 4 ko 3 se bhaag pe 1. Ek number dono kaise? → kabhi nahi. Check: 8 (6 se 2) → 3 se 2; 13 (9 se 4) → 3 se 1 ✅
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🏆 Olympiad Corner
Q1. 7 se 1, 11 se 1, 13 se 1 bache, sabse chhota (1 se bada)?
Solution dekho
LCM 1001 → 1002.
Q2. 3¹⁰ ko 4 se bhaag pe?
Solution dekho
3² = 9 → 1 → 1.
Q3. 1 + 2 + … + 20 ko 7 se?
Solution dekho
210 → 0.
Q4. 100 se kam kitne numbers 4 se 3 chhodte?
Solution dekho
3, 7, …, 99 → 25.
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💗 CharuMam Motivation
Chhotu ne bina gine laddoo ka hisaab pakda. Remainder ka khel bade numbers ko chhota bana deta. Yeh aapki secret mental-maths ki chaabi hai! 🔑
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🟢 Master Recap
- 💡 Chaabi: Har number ko divisor × (poore dabbe) + remainder likh sakte: jaise 7k + 4. Jodne-ghataane-guna mein sirf remainders ka hisaab karo, aur aakhir mein divisor se dobara dabbe banao.
- Number = dk + r (r < d).
- dk + r = d(k + 1) − (d − r).
- Jod/ghataav/guna: sirf remainders.
- Bada remainder → dabba banao; negative → d jodo.
- Same remainder: LCM·k + r.
- Ek kam: number + 1 = LCM.
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👉 Aage Kya?
Remainder ki ghadi se hi 9 aur 11 ke divisibility rules nikalte! Agla post: Post 5 — Divisibility Rule: 9, 3 Aur 11.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬