Post 17.1 mein 2, 5 aur 10 ke rules sirf last digit se the. Par 3 aur 9 ke liye last digit kaam nahi karta: 13 ka last digit 3 hai, phir bhi 13, 3 se divisible nahi! Aaj ka secret: saare digits add karo. Digit sum 3 se divisible, to number bhi 3 se divisible. Digit sum 9 se divisible, to number bhi 9 se. Aur bonus: 6 ka rule = 2 ka rule + 3 ka rule, dono saath.
⏱ Before Timer: Pehle Purane Method Se
Pehle school wala “long division” (ek doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). 351 ÷ 9:
- 3 mein 9 ek baar bhi nahi jaata, isliye pehle do digits 35 lo. 35 mein 9 kitni baar? 9 × 3 = 27 (35 se chhota), 9 × 4 = 36 (bada). To 3. Bacha: 35 − 27 = 8.
- Agla digit 1 neeche utaaro, bache 8 ke saath: 81. 81 mein 9 kitni baar? 9 × 9 = 81. Bacha 0.
- Quotient 39, remainder 0: 351, 9 se divisible (check: 9 × 39 = 351).
9 ki table, subtract karna, digit neeche utaarna. Ab timer wala sawaal isi tarah kijiye.
Ab phone ya clock mein stopwatch (time measure karne wali clock) chalu kijiye. Neeche wale number ko isi long division se 9 se divide kijiye aur remainder dekhiye, copy-pencil ke saath. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
Haan. 5274 ÷ 9 = 586, remainder 0.
Abhi trick nahi dekhni! Post ke end mein After Timer mein yahi sawaal trick se karenge aur dono time ki compare karenge. 😊
— 📌 charumam.com
✅ Pehle Yeh Aata Hai
Chahiye: Post 17.1 ka “divisible” (remainder 0) aur even/odd; Post 16.3 ki baat “har 10 = 9 + 1”; Chapter 2 ka add. Naya: digit sum aur repeated digit sum. Sab neeche ke boxes mein. Naye word:
📖 Word jo is post mein aayenge
- Digit sum: number ke saare digits ka addition: 234 → 2 + 3 + 4 = 9.
- Repeated digit sum: digit sum ka phir digit sum, jab tak ek digit na bache: 4785 → 24 → 6.
- 3 ki table: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 … Inme se koi bhi number 3 se divisible.
- 9 ki table: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 …
- Dono (both): 6 ke rule mein: 2 wala test bhi pass, aur 3 wala test bhi pass.
📏 Bade numbers padhna aur round number mein add karna
Har digit ki ek jagah (place) hoti hai. Right se count karo: pehli jagah Units (ones, 1 wale), uske left mein Tens (10 wale), phir Hundred (100 wale), phir Thousand (1000 wale).
10 ka bundle: 10 units = 1 ten (10). 10 tens = 1 hundred (100). 10 hundreds = 1 thousand (1000). Har jagah apne right wali jagah se 10 times badi hai.
| Number | Kaise padhein | Kis cheez se bana |
|---|---|---|
| 98 | attaanve | 9 tens (90) + 8 units |
| 165 | ek hundred painsath | 1 hundred (100) + 6 tens (60) + 5 units |
| 999 | nau hundred ninyaanve | 9 hundreds (900) + 9 tens (90) + 9 units |
| 1011 | ek thousand gyaarah | 1 thousand (1000) + 0 hundred + 1 ten (10) + 1 unit |
Round number = jiske end mein 0 ho, jaise 10, 100, 1000.
Round number mein chhota number add karna (place box rule): chhote number ko right se likho, yaani units ke neeche units, tens ke neeche tens. Phir har box mein jo hai, woh jawab mein utaar do. Ek digit wala number sirf units ka zero badalta hai, do digit wala units aur tens dono.
- 100 + 65: 65 mein do digit, to 6 tens mein aur 5 units mein: 1|6|5 = 165.
- 100 + 2: 2 sirf units mein, tens ka 0 waise hi rehta hai: 1|0|2 = 102 (12 nahi, 120 nahi!).
- 200 + 6: 6 sirf units mein: 2|0|6 = 206.
- 1000 + 11: 11 mein do digit, to 1 tens mein aur 1 units mein: 1|0|1|1 = 1011.
✅ “Divisible” kya hota hai?
Post 16.1 mein seekha: ÷ matlab equal parts mein divide karna, quotient = har part mein kitna (ya kitne poore group), remainder = jo bach jaaye. Ab ek naya word: divisible.
“24, 4 se divisible hai” matlab 24 ko 4 se divide karo to remainder 0: kuch nahi bachta. 24 ÷ 4 = 6, kyunki 4 × 6 = 24. Isi ko Chapter 16 mein “poora divide” kaha tha. Doosre words mein: 24, 4 ki table mein aata hai (4, 8, 12, 16, 20, 24 …).
“25, 4 se divisible nahi hai”: 25 ÷ 4 → quotient 6, remainder 1, kyunki 25 = 4 × 6 + 1 (pehle multiply: 4 × 6 = 24, phir add: 24 + 1 = 25). Remainder 0 nahi, to divisible nahi. 25, 4 ki table mein nahi aata (24 ke baad seedha 28).
Divisibility rule (ya divisibility test) matlab ek chhota shortcut jo poora division kiye bina bata de ki remainder 0 aayega ya nahi. Is chapter mein yahi shortcuts seekhenge, aur har baar asli division se check bhi karenge.
➕ Digit sum kya hai?
Post 16.3 mein ÷ 9 karte waqt digits add kiye the. Ab us cheez ka name: digit sum = number ke saare digits ko add karna, har digit ko alag number maan kar (place value bhool kar). 4785 ka digit sum: 4 + 7 + 8 + 5 = 24. (4785 ke 4 ka matlab 4000 hai, par digit sum mein bas 4 add karte hain.)
Add karne ka aasaan method (Chapter 2): left se running total rakho: 4, phir 4 + 7 = 11, phir 11 + 8 = 19, phir 19 + 5 = 24.
Repeated digit sum = digit sum ka bhi digit sum, tab tak jab tak ek hi digit na bache. 24 ka digit sum 2 + 4 = 6. To 4785 → 24 → 6. Isse number chhota hota jaata hai aur test aasaan.
🔁 Post 16.3 yaad karo: har 10 = 9 + 1
Post 16.3 ki chaabi: 10 = 9 + 1. Har ek 10 mein se ek poora 9 ka group nikalta hai aur 1 bachta hai. Aur 100 = 99 + 1, jahan 99 = 9 × 11 (9 ke 11 groups), to har 100 se bhi sirf 1 bachta hai. Isliye 43 ÷ 9 ka remainder digits ka sum 4 + 3 = 7 tha. Aaj isi baat se 3 aur 9 ke divisibility rules banenge.
🔁 Chhota sa warm-up
- 2 + 4 + 1 + 2 = ?
- 7 + 4 + 2 + 5 = ?
- 9 × 3 = ?
Answers dekho
- 9
- 18
- 27
9, 18, 27: teeno 9 ki table mein hain. Aaj dekhenge ki jis number ka digit sum 9 ki table mein ho, woh number khud 9 se divisible hota hai!
— 📌 charumam.com
🎬 Story: Sweets Ki Shop
Diwali se pehle halwai uncle ne 2412 laddoo banaye. Har box mein 9 laddoo. Uncle ne poocha: “Saare boxes poore bharenge, ya kuch laddoo bach jaayenge?”
Unka beta calculator dhoondh raha tha. Pados ki Anaya ne kaha: “Digits add karo: 2 + 4 + 1 + 2 = 9. 9, 9 se divisible hai, to laddoo bhi poore boxes mein jaayenge, ek bhi nahi bachega!”
Gin kar dekha: 268 boxes, bilkul poore. 🍬 “Yeh kaise?” Anaya: “Har 10 mein ek 9 aur 1 bachta hai. Bache hue sab mila do, wahi digit sum!” Aaj yahi trick.
— 📌 charumam.com
🪄 Trick Step by Step: Saare Digits Add Karo
💡 Is post ki ek badi baat: 3 ka rule: digit sum 3 se divisible (3, 6, 9, 12, 15 …) → number 3 se divisible. 9 ka rule: digit sum 9 se divisible (9, 18, 27 …) → number 9 se divisible. Digit sum bada ho to uska bhi digit sum lo (repeated digit sum). 6 ka rule: number 2 se bhi divisible (last digit even) aur 3 se bhi (digit sum) → 6 se divisible. Kyunki har 10 = 9 + 1 aur 6 = 2 × 3.
Step 1: Digits add karo
2412 → 2 + 4 + 1 + 2 = 9. (Place value bhool jao: 2 ka matlab 2000 hai, par digit sum mein bas 2 add karo.)
Step 2: 9 ki table mein hai?
9, 9 ki table mein hai → 2412, 9 se divisible. 9, 3 ki table mein bhi hai → 2412, 3 se bhi divisible.
Step 3: Digit sum bada? Phir add karo
98765 → 9 + 8 + 7 + 6 + 5 = 35. 35 table mein hai ya nahi, pakka nahi pata? Phir add karo: 3 + 5 = 8. 8 na 3 ki table mein, na 9 ki → 98765 na 3 se divisible, na 9 se.
Step 4: 6 ke liye do test
5112 ÷ 6? 2 wala test: last digit 2, even ✅. 3 wala test: 5 + 1 + 1 + 2 = 9, 3 ki table mein ✅. Dono pass → 6 se divisible.
Step 5: Check karo (asli division se)
2412 ÷ 9 = 268, kyunki 9 × 268 = 2412 ✅. 5112 ÷ 6 = 852, kyunki 6 × 852 = 5112 ✅. 98765 = 3 × 32921 + 2: remainder 2, divisible nahi ✅.
Aur examples, ek ke baad ek
471 ÷ 3? Digit sum: 4 + 7 + 1 = 12. 12 3 ki table mein hai (3, 6, 9, 12, 15, 18 …) → ✅ divisible. Check: 471 ÷ 3 = 157.
471 ÷ 9? Digit sum: 4 + 7 + 1 = 12. 12 9 ki table mein nahi hai → ❌ divisible nahi. Check: 471 = 9 × 52 + 3.
8001 ÷ 3? Digit sum: 8 + 0 + 0 + 1 = 9. 9 3 ki table mein hai (3, 6, 9, 12, 15, 18 …) → ✅ divisible. Check: 8001 ÷ 3 = 2667.
8001 ÷ 9? Digit sum: 8 + 0 + 0 + 1 = 9. 9 9 ki table mein hai (9, 18, 27, 36 …) → ✅ divisible. Check: 8001 ÷ 9 = 889.
738 ÷ 6? 2 wala test: last digit 8 → even ✅. 3 wala test: 7 + 3 + 8 = 18 → 3 ki table mein ✅. Dono pass → ✅ divisible. Check: 738 ÷ 6 = 123.
742 ÷ 6? 2 wala test: last digit 2 → even ✅. 3 wala test: 7 + 4 + 2 = 13 → 3 ki table mein nahi ❌. Dono pass nahi → ❌ divisible nahi. Check: 742 = 6 × 123 + 4.
Bonus: remainder bhi digit sum se
Post 16.3 mein dekha: ÷ 9 ka remainder = digit sum ka remainder (9 ke groups remove karke). 471 → 4 + 7 + 1 = 12 → 12 − 9 = 3. To 471 ÷ 9 ka remainder 3 (471 = 9 × 52 + 3 ✅). ÷ 3 mein bhi yahi: digit sum 12, 12 ÷ 3 → remainder 0. To 471 ÷ 3 ka remainder 0 (471 ÷ 3 = 157 ✅).
🆘 Atak gaye? Aise socho
- Digit sum: har digit ko alag number maan kar add karo.
- 3: digit sum 3, 6, 9, 12, 15, 18 … 9: digit sum 9, 18, 27 …
- Digit sum bada? Uska bhi digit sum lo (repeated).
- 6: last digit even aur digit sum 3 ki table mein. Ek bhi fail → 6 se nahi.
- Shaq ho to asli division se check.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Har 10 = 9 + 1, har 100 = 99 + 1. 234 = 2 hundreds + 3 tens + 4 units. Har hundred mein 99 (jo 9 × 11 hai, 9 se divisible) aur 1 bachta. Har ten mein 9 aur 1 bachta. Diagram dekho:
2 hundreds se 2 bache, 3 tens se 3 bache, aur 4 units to pure bache. Total bache: 2 + 3 + 4 = 9, yahi digit sum hai! Baaki sab (99 aur 9 wale parts) pehle se 9 se divisible. To agar bache hue (digit sum) 9 se divisible, to 234 khud bhi 9 se divisible.
3 ke liye bhi yahi kyun?
9 = 3 × 3 aur 99 = 3 × 33, dono 3 se bhi divisible. To 3 ke liye bhi sirf bache hue (digit sum) dekhne hain. 6 kyun “2 aur 3 dono”? 6 = 2 × 3. Jo number 2 ki pairs mein bhi poora jaaye aur 3 ke groups mein bhi, woh 6 ke groups mein bhi poora jaata hai. Jaise 12: 2 se ✅, 3 se ✅, 12 ÷ 6 = 2 ✅.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: 7425: 3 se divisible? 9 se?
7425 ÷ 3? Digit sum: 7 + 4 + 2 + 5 = 18. 18 3 ki table mein hai (3, 6, 9, 12, 15, 18 …) → ✅ divisible. Check: 7425 ÷ 3 = 2475.
7425 ÷ 9? Digit sum: 7 + 4 + 2 + 5 = 18. 18 9 ki table mein hai (9, 18, 27, 36 …) → ✅ divisible. Check: 7425 ÷ 9 = 825.
Dono se divisible.
✔️ Normal method se check: 9 × 825 = 7425 ✅
Example 2: 1234: 3 se divisible? 9 se?
1234 ÷ 3? Digit sum: 1 + 2 + 3 + 4 = 10. 10 3 ki table mein nahi hai → ❌ divisible nahi. Check: 1234 = 3 × 411 + 1.
1234 ÷ 9? Digit sum: 1 + 2 + 3 + 4 = 10. 10 9 ki table mein nahi hai → ❌ divisible nahi. Check: 1234 = 9 × 137 + 1.
Kisi se nahi.
✔️ Normal method se check: 1234 = 3 × 411 + 1 ✅
Example 3: 5112 ko 6 se divide karo to remainder 0?
5112 ÷ 6? 2 wala test: last digit 2 → even ✅. 3 wala test: 5 + 1 + 1 + 2 = 9 → 3 ki table mein ✅. Dono pass → ✅ divisible. Check: 5112 ÷ 6 = 852.
✔️ Normal method se check: 6 × 852 = 5112 ✅
Example 4: 2412 laddoo, har box mein 9. Poore boxes bharenge?
2412 ÷ 9? Digit sum: 2 + 4 + 1 + 2 = 9. 9 9 ki table mein hai (9, 18, 27, 36 …) → ✅ divisible. Check: 2412 ÷ 9 = 268.
Haan: 268 boxes, ek bhi laddoo nahi bachega.
✔️ Normal method se check: 9 × 268 = 2412 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
CharuMam ki smart trick: “Bade number ko chhota karo, sach wahi rahega.” 7425 ko check karne ki jagah uska digit sum 18 check karo, aur 18 ki jagah 1 + 8 = 9. Har baar number chhota hota gaya, par “3 se divisible / 9 se divisible” wala jawab nahi badla. Isi soch se Chapter 18 mein addition aur multiplication ke answers bhi 5 second mein check karenge.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| ÷ 3 ya ÷ 9: remainder 0 hai ya nahi | ✅ Haan | Digit sum se. |
| ÷ 9 ya ÷ 3 ka remainder | ✅ Haan | Digit sum ka remainder (9 ya 3 ke groups remove karke). |
| ÷ 6 | ✅ Do test | Last digit even + digit sum 3 ki table mein. |
| ÷ 27 (9 × 3) | ❌ Is rule se nahi | Digit sum 27 ki table mein hona kaafi nahi. Jaise 1899: 1 + 8 + 9 + 9 = 27, par 27 × 70 = 1890, aur 1899 − 1890 = 9, yaani remainder 9: 1899, 27 se divisible nahi. Seedha divide karo. |
| ÷ 7 | ❌ Nahi | Iska koi aasaan digit sum rule nahi. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Last digit dekh liya
Sawaal: 43, 3 se divisible?
❌ Galat: “Last digit 3 hai, to haan.”
✅ Sahi: 3 ke liye digit sum: 4 + 3 = 7, 3 ki table mein nahi → nahi. 43 = 3 × 14 + 1.
Mistake 2: 3 se divisible to 9 se bhi?
Sawaal: 12, 9 se divisible?
❌ Galat: “Digit sum 3 hai, 3 se divisible hai, to 9 se bhi.”
✅ Sahi: 9 ke liye digit sum 9 ki table mein chahiye; 3 nahi hai → nahi. 12 = 9 × 1 + 3. (Reverse sahi hai: 9 se divisible number 3 se bhi divisible.)
Mistake 3: 6 ke liye sirf even dekha
Sawaal: 14, 6 se divisible?
❌ Galat: “14 even hai, to haan.”
✅ Sahi: 3 wala test bhi: 1 + 4 = 5, fail → nahi. 14 = 6 × 2 + 2.
Mistake 4: Digit sum mein place value laga di
Sawaal: 234 ka digit sum
❌ Galat: “200 + 30 + 4 = 234” (yeh to number hi wapas aa gaya).
✅ Sahi: digits ko akele numbers maan kar: 2 + 3 + 4 = 9.
🧠 Rule of thumb: 3 aur 9: digits add karo, digit sum 3 / 9 ki table mein ho to divisible. 6: last digit even AUR digit sum 3 ki table mein.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: “Haan/Nahi” aur kuch remainder wale. Target: 3 minute.
- 2412: 9 se divisible?
- 471: 3 se divisible?
- 471: 9 se divisible?
- 1234: 3 se divisible?
- 8001: 9 se divisible?
- 738: 6 se divisible?
- 742: 6 se divisible?
- 5555: 3 se divisible?
- 9990: 9 se divisible?
- 1236: 6 se divisible?
- 98765: 3 se divisible?
- 111: 3 se divisible?
- 471 ÷ 9 ka remainder = ?
- 1000 ÷ 3 ka remainder = ?
- 2024 ÷ 9 ka remainder = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- Haan (digit sum 9)
- Haan (digit sum 12)
- Nahi (digit sum 12)
- Nahi (digit sum 10)
- Haan (digit sum 9)
- Haan (last digit 8, digit sum 18)
- Nahi (last digit 2, digit sum 13)
- Nahi (digit sum 20)
- Haan (digit sum 27)
- Haan (last digit 6, digit sum 12)
- Nahi (digit sum 35)
- Haan (digit sum 3)
- 3
- 1
- 8
Kaise kiya (kuch): 9990: 9 + 9 + 9 + 0 = 27 → 9 ki table mein → haan. 742: even ✅, par 7 + 4 + 2 = 13 → 3 se nahi → 6 se nahi. 2024 ÷ 9: 2 + 0 + 2 + 4 = 8 → remainder 8.
— 📌 charumam.com
⏰ One-Minute Challenge
Sirf 1 minute (60 second). Timer lagao, aur jitne ho sakein utne sawaal mann mein (bina copy ke) karo. Sirf jawab likhna hai.
- 12 ÷ 3 ka remainder = ?
- 25 ÷ 3 ka remainder = ?
- 81 ÷ 9 ka remainder = ?
- 100 ÷ 9 ka remainder = ?
- 123 ÷ 3 ka remainder = ?
- 222 ÷ 9 ka remainder = ?
- 505 ÷ 3 ka remainder = ?
- 1111 ÷ 9 ka remainder = ?
- 4004 ÷ 3 ka remainder = ?
- 9876 ÷ 9 ka remainder = ?
| Rating | Kitne sahi, kitne time mein |
|---|---|
| 🌱 Beginner | 60 second mein 5 se 7 sahi. Badhiya start! |
| 🚀 Pro | 60 second mein 8 ya 9 sahi |
| 🏆 Champion | 10 mein 10 sahi, 60 second ke andar |
Answers dekho
- 0
- 1
- 0
- 1
- 0
- 6
- 1
- 4
- 2
- 3
Dhyaan: 222 ÷ 9: 2 + 2 + 2 = 6 → remainder 6. 505 ÷ 3: 5 + 0 + 5 = 10, 10 ÷ 3 → remainder 1. 9876 ÷ 9: 9 + 8 + 7 + 6 = 30 → 3 + 0 = 3 → remainder 3.
5 se kam sahi? Koi baat nahi! Trick wala section ek baar aur padho aur kal phir try karo. Har day thoda better, bas yahi chahiye. 💪
— 📌 charumam.com
📝 10 MCQs (Har Option Ka Reason)
MCQ matlab ek sawaal ke chaar jawab (options) diye hote hain, unme se ek sahi chunna hai.
Q1. 2412, 9 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (b) Haan, digit sum 9 hai
- (a) galat ❌: Kisi ek digit ka even hona 9 ke liye kuch nahi batata.
- (b) sahi ✅: 2 + 4 + 1 + 2 = 9 → 9 ki table mein → divisible (2412 ÷ 9 = 268).
- (c) galat ❌: 9 ke liye last digit nahi, digit sum dekhte hain.
- (d) galat ❌: Even hona sirf 2 ka test hai, 9 ka nahi.
Q2. Kaunsa number 3 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (a) 471
- (a) sahi ✅: 4 + 7 + 1 = 12 → 3 ki table mein (471 ÷ 3 = 157).
- (b) galat ❌: 4 + 7 + 2 = 13 → nahi (remainder 1).
- (c) galat ❌: 4 + 7 + 3 = 14 → nahi (remainder 2).
- (d) galat ❌: 4 + 7 + 5 = 16 → nahi (remainder 1).
Q3. 98765 ka repeated digit sum?
Jawab aur reason dekho
Sahi jawab: (d) 8
- (a) galat ❌: 35 pehla digit sum hai; repeated mein ek digit tak jaate hain.
- (b) galat ❌: 5 sirf last digit hai.
- (c) galat ❌: 11 kahin nahi aata; 35 ka digit sum 8 hai.
- (d) sahi ✅: 9 + 8 + 7 + 6 + 5 = 35, phir 3 + 5 = 8.
Q4. 6 se divisible hone ke liye kya chahiye?
Jawab aur reason dekho
Sahi jawab: (c) 2 aur 3 dono ke test pass
- (a) galat ❌: 15 ka digit sum 1 + 5 = 6 hai, par 15 odd hai, isliye 6 se divisible nahi.
- (b) galat ❌: Even hona half kaam; 3 wala test bhi chahiye (14 even par 6 se nahi).
- (c) sahi ✅: 6 = 2 × 3, isliye dono test.
- (d) galat ❌: 16 ka last digit 6, par 16 = 6 × 2 + 4: nahi.
Q5. 742, 6 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (b) Nahi, digit sum 13 hai
- (a) galat ❌: 742 even hai (last digit 2).
- (b) sahi ✅: Even ✅, par 7 + 4 + 2 = 13 → 3 se nahi, to 6 se nahi.
- (c) galat ❌: Sirf 2 wala test pass hua; 3 wala fail.
- (d) galat ❌: Digit sum mein saare digits add karne hote hain: 13.
Q6. 471 ÷ 9 ka remainder?
Jawab aur reason dekho
Sahi jawab: (a) 3
- (a) sahi ✅: 4 + 7 + 1 = 12, 12 − 9 = 3. Check: 9 × 52 + 3 = 471.
- (b) galat ❌: 0 tab hota jab digit sum 9 ki table mein hota.
- (c) galat ❌: 12 mein abhi ek 9 aur hai; remainder 9 se chhota hota hai.
- (d) galat ❌: 1 sirf last digit hai.
Q7. Kaunsa statement sahi hai?
Jawab aur reason dekho
Sahi jawab: (d) 9 se divisible number, 3 se bhi divisible
- (a) galat ❌: 12, 3 se divisible hai par 9 se nahi.
- (b) galat ❌: 6 se divisible number 2 se bhi divisible hai, yaani hamesha even.
- (c) galat ❌: 12 ka digit sum 3 hai, par 12, 9 se divisible nahi.
- (d) sahi ✅: 9 = 3 × 3, to 9 ke groups, 3 ke groups mein bhi toot jaate hain.
Q8. Digit sum rule 9 ke liye kyun chalta hai?
Jawab aur reason dekho
Sahi jawab: (c) Har 10 = 9 + 1 aur har 100 = 99 + 1
- (a) galat ❌: 12 mein 9 nahi hai.
- (b) galat ❌: Odd hona kuch nahi batata; 7 bhi odd hai par uska aisa rule nahi.
- (c) sahi ✅: Har ten aur hundred se 1 bachta hai; bache hue ka total = digit sum.
- (d) galat ❌: Sabse bada digit hona reason nahi.
Q9. 5□2 (□ ek digit) ko 9 se divisible banana hai. □ = ?
Jawab aur reason dekho
Sahi jawab: (b) 2
- (a) galat ❌: 5 + 9 + 2 = 16 → nahi.
- (b) sahi ✅: 5 + 2 + 2 = 9 → 522 ÷ 9 = 58.
- (c) galat ❌: 5 + 0 + 2 = 7 → nahi.
- (d) galat ❌: 5 + 7 + 2 = 14 → nahi.
Q10. 1000 ÷ 3 ka remainder?
Jawab aur reason dekho
Sahi jawab: (a) 1
- (a) sahi ✅: 1 + 0 + 0 + 0 = 1 → remainder 1 (1000 = 3 × 333 + 1).
- (b) galat ❌: 0 tab hota jab digit sum 3 ki table mein hota.
- (c) galat ❌: Remainder 3 se chhota hota hai (0, 1 ya 2).
- (d) galat ❌: 10 kahin nahi aata; digit sum 1 hai.
— 📌 charumam.com
🧠 Challenge: HOTS Aur Olympiad Style
HOTS matlab Higher Order Thinking Skills, yaani aise sawaal jinme sirf trick nahi, thoda mind bhi lagana padta hai. Olympiad ek maths competition hota hai.
Q1. Sabse chhota 4-digit number kaunsa hai jo 9 se divisible ho?
Solution dekho
Sabse chhota 4-digit number 1000, digit sum 1. 9 tak pahunchne ke liye 8 aur chahiye, aur sabse chhota method last digit badhana: 1008 (1 + 0 + 0 + 8 = 9).
Check: 1008 ÷ 9 = 112 ✅
Q2. 7425 ke digits ko kisi bhi order mein rakho (jaise 5274, 2457). Kya har naya number 9 se divisible hoga?
Solution dekho
Haan! Digit sum order se nahi badalta: 7 + 4 + 2 + 5 = 18, hamesha. To har arrangement 9 se divisible. Jaise 5274 ÷ 9 = 586 ✅ aur 2457 ÷ 9 = 273 ✅.
Q3. 111111111 (nau baar 1), 9 se divisible hai?
Solution dekho
Digit sum: nau baar 1 = 9 → haan, 9 se divisible. (111111111 ÷ 9 = 12345679.) Aur 111 (teen baar 1): digit sum 3 → 3 se divisible: 111 ÷ 3 = 37 ✅.
Q4. Ek 3-digit number 6 se divisible hai aur uske digits 4, □, 6 hain (4□6). □ ke kitne options?
Solution dekho
Last digit 6 even, to 2 wala test hamesha pass. 3 wala test: 4 + □ + 6 = 10 + □, 3 ki table mein chahiye: 12, 15, 18 → □ = 2, 5, 8. Teen options: 426, 456, 486.
Check: 426 ÷ 6 = 71, 456 ÷ 6 = 76, 486 ÷ 6 = 81 ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | 3, 9, 6 ke rules, missing digit (5□2), “kaunsa number divisible hai”. |
| Maths Olympiad | Digit rearrange wale sawaal, sabse chhota/bada number jo divisible ho. |
| NTSE / MAT | Number series aur tez Simplification. |
| SSC / Bank (aage) | Number system: remainder aur divisibility ke sawaal har exam mein. |
Shabdon ka matlab: Olympiad = school ke students ki maths competition. NTSE = Class 10 ke students ka ek national scholarship exam (jisme achhe marks pe padhai ke liye paise milte hain). MAT = Mental Ability Test, yaani sochne-samajhne ka exam (NTSE ka ek part). SSC / Bank exam = badon ke government job ke exam. Simplification = bade calculation ko chhota karke jawab nikaalna. Number series = numbers ki ek line jo kisi rule se aage increase hoti hai, aur agla number batana hota hai. Jaise 5, 10, 15, 20, … mein har baar 5 jud raha hai, to agla 25. Class 6–8 = school ki chhathi se aathvi class. Number system = numbers ka chapter: even/odd, divisibility, remainder jaise rules aur unke sawaal.
— 📌 charumam.com
👩🏫 CharuMam Ka Tip
Mantra: “3 aur 9 ke liye digits add karo; 6 ke liye even bhi dekho.” Mobile number, car ka number: digits add karke dekho 3 ya 9 se divisible hai ya nahi. Agle post mein 4, 8 aur 11: last 2 digits, last 3 digits aur ek mazedaar “alternate” sum! ➕
— 📌 charumam.com
⏱ After Timer: Ab Trick Se!
Wahi sawaal jo shuru mein kiya tha. Stopwatch phir chalu kijiye, aur is baar aaj ki trick se kijiye:
Trick se kaise karein (dekho)
Digits add karo: 5 + 2 + 7 + 4 = 18. 18, 9 ki table mein hai (9, 18, 27 …) → divisible ✅. Na table, na subtract karna!
| Before time | After time | Kitna tez? |
|---|---|---|
| ____ second | ____ second | Before − After = ____ second bache! |
Pehli baar mein time zyada bhi lag sakta hai, ghabraiye nahi! Nayi trick ko 3 se 4 days ki practice chahiye. Phir yeh purane method se kaafi tez ho jaati hai. 🚀
— 📌 charumam.com
🤖 Aur Practice: AI Tutor Se Poochho
Neeche wali line copy kijiye aur CharuMam AI Tutor (tutor.charumam.com) mein paste kar dijiye. Tutor aapko naye sawaal dega aur mistake hone pe samjhayega:
— 📌 charumam.com
❓ FAQs (Students Ke Asli Sawaal)
1. Digit sum kitni baar repeat kar sakte hain?
Jitni baar chaho, jab tak ek digit na bache. 3 aur 9 ka jawab nahi badalta.
2. Remainder ke liye bhi digit sum chalega?
÷ 9 aur ÷ 3 ke liye haan: digit sum ka remainder hi number ka remainder. ÷ 6 ke liye nahi.
3. Agar digit sum 9 hai, to number 3 se bhi divisible?
Haan, kyunki 9, 3 ki table mein hai.
4. 0 wale digits ka kya?
0 add karne se kuch nahi badalta: 9090 → 9 + 0 + 9 + 0 = 18.
5. Agle post mein kya hai?
Post 17.3: 4 (last 2 digits), 8 (last 3 digits) aur 11 (alternate digits ka difference).
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Digit sum = saare digits ka addition; repeated digit sum = ek digit tak.
- 3: digit sum 3 ki table mein. 9: digit sum 9 ki table mein.
- 6: last digit even AUR digit sum 3 ki table mein.
- Kyun: har 10 = 9 + 1, har 100 = 99 + 1; bache hue = digit sum.
- ÷ 9 aur ÷ 3 ka remainder bhi digit sum se.
— charumam.com 🎯
👉 Aage Kya?
Ab Post 17.3: 4, 8 aur 11. 4 ke liye sirf last 2 digits, 8 ke liye last 3 digits (kyunki 100 = 4 × 25 aur 1000 = 8 × 125), aur 11 ke liye digits pe + aur − baari-baari: 7392, 11 se divisible!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
💛 Post Achha Laga? Like, Share, Subscribe, Follow!
Agar aaj kuch naya samajh aaya, to post ko like kijiye, apne doston aur class ke group mein share kijiye, aur daily nayi trick ke liye humein subscribe aur follow kijiye! 🙏
- 🌐 Website: charumam.com
- 🤖 AI Tutor (doubt poochho, 24×7): tutor.charumam.com
- ▶️ YouTube (subscribe): @charumamofficial
- 📸 Instagram (follow): @charumamofficial
- ✈️ Telegram: t.me/charumam
- 💬 WhatsApp Channel: CharuMam WhatsApp Channel
- 𝕏 X (follow): @CharuMamHQ
— 📌 charumam.com
