Chapter 17 ka last post! 🏆 Aaj 17.1–17.3 ka mix, soch wale sawaal: kaunsa rule kab, missing digit (37□ ko 9 se divisible banao), combined rules (12 = 3 aur 4 dono, 15 = 3 aur 5, 18 = 2 aur 9), remainder jaldi, aur dhokhe jaise “4 aur 2 se divisible, to 8 se bhi?” Har jawab ka check asli division se.
⏱ Before Timer: Pehle Purane Method Se
Pehle school wala “long division” (doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). 1764 ÷ 12:
- 1 mein 12 nahi jaata, to pehle do digits 17. 17 mein 12: 1 baar (12). Upar 1. Bacha 17 − 12 = 5.
- Agla digit 6 neeche: 56. 56 mein 12: 4 baar (12 × 4 = 48). Upar 4. Bacha 56 − 48 = 8.
- Agla digit 4 neeche: 84. 84 mein 12: 7 baar (12 × 7 = 84). Upar 7. Bacha 0.
- Quotient 147, remainder 0 → 1764, 12 se divisible (check: 12 × 147 = 1764).
12 ki table, subtract, digit neeche: kaafi kaam. 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 12 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. 4356 ÷ 12 = 363, 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: Chapter 17 ke teeno post, Post 16.1 ka ÷ aur remainder, aur Post 16.3 ka “÷ 9 = digits add karo”. Yaad ke liye boxes neeche. Naye word:
📖 Word jo is post mein aayenge
- Divisor: jis number se divide karte hain: 4356 ÷ 12 mein divisor 12.
- Combined rule: do chhote rules ek saath: 12 ke liye 3 ka test aur 4 ka test.
- Common divisor: jo number do numbers ko dono ko poora divide kare.
- Missing digit (□): number mein ek digit ki jagah empty box; usme sahi digit dhoondhna hai.
- Dhokha (trap): aisa sawaal jo dekhne mein aasaan lage par galat jawab ki taraf le jaaye.
📏 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.
➗ Division kya hai? ÷, quotient aur remainder
Post 16.1 mein seekha tha: ÷ (divide ka sign) matlab equal parts mein divide karna. “12 ÷ 4” padho “12 divided by 4” ya “12 ko 4 se divide”: 12 laddoo 4 students mein equal divide karo, har ek ko 3. Ise aise bhi soch sakte ho: 12 mein 4-4 ke kitne group bante hain? 3 group. Dono ka jawab ek hi: 12 ÷ 4 = 3, kyunki 4 × 3 = 12.
Quotient = division ka jawab: har ek ko kitna mila, ya kitne poore group bane. Remainder = jo aakhir mein bach gaya, jisse ek aur poora group nahi ban sakta. 47 laddoo, 5-5 ke boxes: 9 boxes bharte hain (5 × 9 = 45), aur 2 laddoo bach jaate hain. To quotient 9 (9 poore boxes) aur remainder 2 (2 bache laddoo). Hum likhenge: 47 ÷ 5 → quotient 9, remainder 2. (Remainder hamesha divide karne wale number se chhota hota hai: 5 se divide mein remainder 0, 1, 2, 3 ya 4.)
Check ka rule: divide karne wala × quotient + remainder = asli number. Is line mein pehle multiply, phir add: 47 = 5 × 9 + 2, kyunki 5 × 9 = 45 aur 45 + 2 = 47 ✅. (Agar pehle 9 + 2 = 11 add karte aur phir 5 × 11 = 55 karte, to galat hota.)
Poora divide: jab remainder 0 ho, to kehte hain number “poora divide” ho gaya: 45 ÷ 5 = 9, kuch nahi bacha.
✅ “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.
🤝 “Common divisor” ka matlab
Do numbers ka common divisor = aisa number jo dono ko poora divide kare. 4 aur 6: dono 2 se divisible, to 2 common divisor. 3 aur 4: dono ko poora divide karne wala sirf 1, to “1 ke alawa koi common divisor nahi”.
Aaj ka bada rule: agar 12 = 3 × 4, aur 3, 4 ka 1 ke alawa koi common divisor nahi, to “12 se divisible” = “3 se bhi divisible aur 4 se bhi divisible”. Yahi baat 15 = 3 × 5 aur 18 = 2 × 9 pe bhi. 12 = 3 × 4 ✅ (3 aur 4 ka sirf 1). Par 12 = 2 × 6 ❌ (2 aur 6 dono 2 se divisible), isliye 12 ke liye “2 aur 6” wala test galat ho sakta hai.
🔁 Chhota sa warm-up
- 3 × 4 = ?
- 3 × 5 = ?
- 2 × 9 = ?
Answers dekho
- 12
- 15
- 18
12, 15, 18: har ek do aise numbers ka multiply hai jinka 1 ke alawa koi common divisor nahi. Isliye inke rules do chhote rules dono saath mila kar bante hain!
— 📌 charumam.com
🎬 Story: Annual Day Ke Laddoo
Annual day pe 1080 laddoo aaye. Principal ma’am ne kaha: “Teams banengi: kabhi 12 students ki, kabhi 15 ki, kabhi 18 ki. Har baar laddoo equal divide hone chahiye, ek bhi na bache!”
Kabir ne turant kaha: “1080: last digit 0, digit sum 1 + 0 + 8 + 0 = 9, last 2 digits 80 = 4 × 20. To 3, 4, 5, 9, 2 sab pass! 12 (3 aur 4), 15 (3 aur 5), 18 (2 aur 9): teeno chalenge!”
Ma’am ne check kiya: 90, 72, 60 laddoo har team ko, bilkul equal. 🎉 Aaj aise hi mix sawaal!
— 📌 charumam.com
🪄 Trick Step by Step: Pehle Identify karo, Phir Rule Chuno
💡 Is post ki ek badi baat: Pehle divisor dekho. 2, 5, 10 → last digit. 4 → last 2 digits. 8 → last 3 digits. 3, 9 → digit sum. 11 → alternate sums. Bade divisor (12, 15, 18, 24, 36, 45) → do chhote numbers mein break karo jinka 1 ke alawa koi common divisor nahi (12 = 3 × 4, 15 = 3 × 5, 18 = 2 × 9), dono tests pass hone chahiye. Missing digit mein har digit 0 se 9 try karo (ya digit sum se seedha socho). Aakhir mein check.
Step 1: Divisor identify karo
Kis se divisible poocha hai: 2, 3, 4, 5, 6, 8, 9, 10, 11, ya bada jaise 12? Diagram dekho.
Step 2: Bada divisor? Break karo
12 = 3 × 4, 15 = 3 × 5, 18 = 2 × 9, 24 = 3 × 8, 36 = 4 × 9, 45 = 5 × 9. Har pair mein dono numbers ka 1 ke alawa koi common divisor nahi.
Step 3: Dono tests lagao
1080 ÷ 18? 2 wala test: last digit 0, even ✅. 9 wala test: 1 + 0 + 8 + 0 = 9 ✅ → divisible.
Step 4: Missing digit? Rule reverse chalao
37□ ko 9 se divisible: 3 + 7 = 10, aur digit sum 9 ki table mein chahiye: 18. To □ = 18 − 10 = 8 (378).
Yaad rakho: □ mein sirf ek digit (0 se 9) aa sakta hai, to 3 + 7 + □ = 10 + □ sirf 10 se 19 tak ho sakta hai. 9 ki table mein: 9 (10 se chhota, nahi ho sakta), 18 (□ = 8 ✅), 27 (□ = 27 − 10 = 17 chahiye, par 17 ek digit nahi ❌). Isliye sirf □ = 8.
Step 5: Check karo
1080 ÷ 18 = 60, kyunki 18 × 60 = 1080 ✅. 378 ÷ 9 = 42, kyunki 9 × 42 = 378 ✅.
Type 1: Kaunsa rule kab?
| Sawaal | Rule | Jawab |
|---|---|---|
| 4592 ÷ 8? | last 3 digits 592 → 296 → 148 → 74 | Haan |
| 4592 ÷ 3? | digit sum 4 + 5 + 9 + 2 = 20 | Nahi |
| 4592 ÷ 11? | (2 + 5) aur (9 + 4): difference 13 − 7 = 6 | Nahi |
| 4592 ÷ 5? | last digit 2 | Nahi |
Type 2: Missing digit
4□52 ko 3 se divisible: 4 + 5 + 2 = 11. □ mein sirf ek digit (0 se 9) aa sakta hai, to 11 + □ sirf 11 se 20 tak ho sakta hai. Is range mein 3 ki table: 12, 15, 18. To □ = 1, 4 ya 7 (teen jawab!): 4152, 4452, 4752. Agla 21 hota, par uske liye □ = 21 − 11 = 10 chahiye, jo ek digit nahi ❌. Check: 4152 ÷ 3 = 1384, 4452 ÷ 3 = 1484, 4752 ÷ 3 = 1584 ✅.
5□4 ko 12 se divisible: 4 wala test: last 2 digits □4 → 04, 24, 44, 64, 84 pass (14, 34, 54, 74, 94 fail). 3 wala test: 5 + □ + 4 = 9 + □ → □ = 0, 3, 6, 9. Dono mein common: □ = 0 ya 6 → 504, 564. Check: 504 ÷ 12 = 42, 564 ÷ 12 = 47 ✅.
Type 3: Combined rules
1080 ÷ 12? 12 = 3 × 4 (dono ka 1 ke alawa koi common divisor nahi). 3 wala test: 1 + 0 + 8 + 0 = 9, 3 ki table mein ✅; 4 wala test: last 2 digits 80 = 4 × 20 ✅ → ✅ divisible. Check: 1080 ÷ 12 = 90.
1080 ÷ 15? 15 = 3 × 5 (dono ka 1 ke alawa koi common divisor nahi). 3 wala test: 1 + 0 + 8 + 0 = 9, 3 ki table mein ✅; 5 wala test: last digit 0 ✅ → ✅ divisible. Check: 1080 ÷ 15 = 72.
2345 ÷ 15? 15 = 3 × 5 (dono ka 1 ke alawa koi common divisor nahi). 3 wala test: 2 + 3 + 4 + 5 = 14, 3 ki table mein nahi ❌; 5 wala test: last digit 5 ✅ → ❌ divisible nahi. Check: 2345 = 15 × 156 + 5.
4356 ÷ 12? 12 = 3 × 4 (dono ka 1 ke alawa koi common divisor nahi). 3 wala test: 4 + 3 + 5 + 6 = 18, 3 ki table mein ✅; 4 wala test: last 2 digits 56 = 4 × 14 ✅ → ✅ divisible. Check: 4356 ÷ 12 = 363.
Type 4: Remainder jaldi
🧮 Pehle samjho: digit sum se remainder kyun milta hai?
÷ 9 (aur ÷ 3) ke liye: digit sum mein se 9-9 nikaalne ke baad jo bachta hai, wahi remainder hai. Example: 47 laddoo, 9-9 ke boxes mein. 4 + 7 = 11. 11 mein ek 9 aur hai: 11 − 9 = 2. Asli division: 47 = 9 × 5 + 2, remainder bhi 2. Dono same! Kyun? Har 10 = 9 + 1 (Post 16.3): 47 = 4 tens + 7 = (4 × 9 + 4) + 7. 9 ke groups remove kar do, bachte hain 4 + 7 = 11 = 9 + 2, yaani 2.
Ek hi method: digits add karte jao jab tak ek digit na bache; woh digit hi ÷ 9 ka remainder (agar 9 aaye to remainder 0). “9 subtract karo” aur “digits phir add karo” ka jawab hamesha same aata hai: 10 → 1 + 0 = 1, aur 10 − 9 = 1. 21 → 2 + 1 = 3, aur 21 − 9 − 9 = 3. (Do-digit number ke digits add karna bhi har ten mein se ek 9 remove karna hi hai.)
÷ 3 ke liye: digit sum ko 3 se divide karo, jo bache woh remainder. 47 → 11 = 3 × 3 + 2 → remainder 2. Check: 47 = 3 × 15 + 2 ✅.
÷ 4 ke liye: sirf last 2 digits ka remainder, kyunki 100 = 4 × 25 (hundreds hamesha 4 se poore). 347 → 47 = 4 × 11 + 3 → remainder 3. Check: 347 = 4 × 86 + 3 ✅.
12345 ÷ 9: 1 + 2 + 3 + 4 + 5 = 15 → 1 + 5 = 6 (ya 15 − 9 = 6, same) → remainder 6. 12345 ÷ 4: last 2 digits 45 = 4 × 11 + 1 → remainder 1. 12345 ÷ 5: last digit 5 → remainder 0. Check: 12345 = 9 × 1371 + 6 ✅.
🆘 Atak gaye? Aise socho
- Divisor identify karo (diagram ki list).
- Bada divisor: do aise numbers mein break karo jinka 1 ke alawa koi common divisor nahi.
- Dono tests pass hone chahiye; ek bhi fail to nahi.
- Missing digit: kai jawab ho sakte hain; sab likho.
- Check: asli division (divisor × quotient + remainder).
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
12 ke liye 3 aur 4 dono kyun? Tables dekho.
3 ki table: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 …
4 ki table: 4, 8, 12, 16, 20, 24, 28, 32, 36 …
Dono mein common: 12, 24, 36 …, yaani bilkul 12 ki table. To jo number 3 se bhi divisible ho aur 4 se bhi, woh 12 ki table mein hai, yaani 12 se divisible.
24 laddoo ki picture: 24 laddoo 3-3 ke 8 groups mein poore jaate hain (3 × 8 = 24), aur 4-4 ke 6 groups mein bhi (4 × 6 = 24). Isliye 12-12 ke 2 boxes mein bhi poore jaate hain (12 × 2 = 24).
“Tabhi pakka” ka matlab: yeh tabhi chalta hai jab dono numbers ka 1 ke alawa koi common divisor na ho. 3 aur 4 ka common sirf 1 hai, isliye unka pehla common number 3 × 4 = 12 hai, aur saare common numbers 12 ki table hain.
8 ke liye 4 aur 2 kyun nahi? 2 ki table: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 … 4 ki table: 4, 8, 12, 16, 20 … Common: 4, 8, 12, 16, 20 …, yaani 4 ki table, 8 ki table nahi. 12 aur 20 dono tables mein hain, par 8 ki table mein nahi (12 = 8 × 1 + 4, 20 = 8 × 2 + 4) ❌. Kyunki 4 aur 2 ka common divisor 2 hai: 4 ki table already 2 ki table ke andar hai, to 2 wala test kuch naya nahi batata. Isliye pair chunte waqt common divisor dekhna zaroori.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: 1080 laddoo, 18 students ki team: equal divide honge?
1080 ÷ 18? 18 = 2 × 9 (dono ka 1 ke alawa koi common divisor nahi). 2 wala test: last digit 0 even ✅; 9 wala test: 1 + 0 + 8 + 0 = 9, 9 ki table mein ✅ → ✅ divisible. Check: 1080 ÷ 18 = 60.
Haan: har student ko 60 laddoo.
✔️ Normal method se check: 18 × 60 = 1080 ✅
Example 2: 37□ ko 9 se divisible banao. □ = ?
3 + 7 = 10. 9 ki table mein 10 se agla 18. □ = 18 − 10 = 8. Number 378.
378 ÷ 9? Digit sum: 3 + 7 + 8 = 18. 18 9 ki table mein hai (9, 18, 27, 36 …) → ✅ divisible. Check: 378 ÷ 9 = 42.
✔️ Normal method se check: 9 × 42 = 378 ✅
Example 3: ₹ 2340 ko 12 students mein equal divide kar sakte hain (kuch na bache)?
2340 ÷ 12? 12 = 3 × 4 (dono ka 1 ke alawa koi common divisor nahi). 3 wala test: 2 + 3 + 4 + 0 = 9, 3 ki table mein ✅; 4 wala test: last 2 digits 40 = 4 × 10 ✅ → ✅ divisible. Check: 2340 ÷ 12 = 195.
Haan: har student ko ₹ 195.
✔️ Normal method se check: 12 × 195 = 2340 ✅
Example 4: 7392: 2, 3, 4, 8, 11 mein se kis-kis se divisible?
7392 ÷ 2? Last digit (units) 2: 2 even hai (0, 2, 4, 6, 8 mein se) → ✅ divisible. Check: 7392 ÷ 2 = 3696.
7392 ÷ 3? Digit sum: 7 + 3 + 9 + 2 = 21. 21 3 ki table mein hai (3, 6, 9, 12, 15, 18 …) → ✅ divisible. Check: 7392 ÷ 3 = 2464.
7392 ÷ 4? Last 2 digits: 92. 92 ÷ 4: 92 = 4 × 23 → ✅ divisible. Check: 7392 ÷ 4 = 1848.
7392 ÷ 8? Last 3 digits: 392. 392 ÷ 8: 392 = 8 × 49 → ✅ divisible. Check: 7392 ÷ 8 = 924.
7392 ÷ 11? Right se digits: 1st, 3rd, 5th … place wale: 2 + 3 = 5. 2nd, 4th … place wale: 9 + 7 = 16. Difference (bada − chhota): 16 − 5 = 11. 11 11 ki table mein hai (11, 22, 33 …) → ✅ divisible. Check: 7392 ÷ 11 = 672.
✔️ Normal method se check: 7392 ÷ 8 = 924, 7392 ÷ 11 = 672, 7392 ÷ 3 = 2464 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
CharuMam ki smart trick: “Bade ko break karo, chhote ko jeeto.” 18 ka koi seedha rule yaad nahi? Koi baat nahi: 18 = 2 × 9, aur 2 aur 9 ke rules aate hain. Bas pair aisi chuno jinka 1 ke alawa koi common divisor na ho. Aur missing digit mein rule ko reverse chalao: “digit sum kitna chahiye?” se seedha □ mil jaata hai.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| Divisor 2, 3, 4, 5, 6, 8, 9, 10, 11 | ✅ Haan | Seedha rule. |
| Divisor 12, 15, 18, 24, 36, 45 | ✅ Haan | Do tests (3 × 4, 3 × 5, 2 × 9, 3 × 8, 4 × 9, 5 × 9). |
| Missing digit wala sawaal | ✅ Haan | Rule reverse chalao; saare jawab likho. |
| Quotient bhi chahiye | ⚠️ Rule + division | Rule haan/nahi batata; quotient ke liye divide karo. |
| Divisor 7, 13, 17 | ❌ Nahi | Inke aasaan rules nahi; seedha divide karo. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: 4 aur 2 se divisible, to 8 se bhi?
Sawaal: 12, 8 se divisible?
❌ Galat: “4 se ✅, 2 se ✅, 4 × 2 = 8, to haan.”
✅ Sahi: 4 aur 2 ka common divisor 2 hai, yeh pair nahi chalti. 12 = 8 × 1 + 4 → nahi.
Mistake 2: 6 aur 4 se divisible, to 24 se bhi?
Sawaal: 36, 24 se divisible?
❌ Galat: “6 se ✅, 4 se ✅, 6 × 4 = 24, to haan.”
✅ Sahi: 24 = 3 × 8 lo (3 aur 8 ka sirf 1). 8 wala test: 36 → 18 → 9: sirf 2 half hue, 9 odd hai aur 3rd half nahi ho sakta ❌ → nahi. 36 = 24 × 1 + 12.
Mistake 3: Missing digit ka ek hi jawab
Sawaal: 4□52, 3 se divisible: □ = ?
❌ Galat: “Sirf 1.”
✅ Sahi: 4 + □ + 5 + 2 = 11 + □ → 12, 15, 18 → □ = 1, 4, 7 (teeno).
Mistake 4: 9 se divisible, to 18 se bhi?
Sawaal: 27, 18 se divisible?
❌ Galat: “Digit sum 9, to 18 se bhi.”
✅ Sahi: 18 = 2 × 9; 27 odd hai, 2 wala test fail → nahi. 27 = 18 × 1 + 9.
🧠 Rule of thumb: Bada divisor = do aise chhote numbers ka multiply jinka 1 ke alawa koi common divisor nahi; dono tests pass → divisible. Missing digit: rule reverse chalao, saare jawab likho.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 mix sawaal. Target: 5 minute.
- 1080: 12 se divisible?
- 1080: 15 se divisible?
- 1080: 18 se divisible?
- 2345: 15 se divisible?
- 4356: 12 se divisible?
- 5112: 18 se divisible?
- 7200: 24 se divisible?
- 8136: 36 se divisible?
- 1234: 12 se divisible?
- 4592: 8 se divisible?
- 4592: 11 se divisible?
- 37□: 9 se divisible, □ = ?
- 4□52: 3 se divisible, □ = ?
- 5□4: 12 se divisible, □ = ?
- 12345 ÷ 9 ka remainder = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- Haan (3: ✅, 4: ✅)
- Haan (3: ✅, 5: ✅)
- Haan (2: ✅, 9: ✅)
- Nahi (3: ❌, 5: ✅)
- Haan (3: ✅, 4: ✅)
- Haan (2: ✅, 9: ✅)
- Haan (3: ✅, 8: ✅)
- Haan (4: ✅, 9: ✅)
- Nahi (3: ❌, 4: ❌)
- Haan (last 3 digits 592)
- Nahi (difference 6)
- 8
- 1, 4, 7
- 0, 6
- 6
Kaise kiya (kuch): 7200 ÷ 24: 24 = 3 × 8; 3 wala test: 7 + 2 + 0 + 0 = 9, 3 ki table mein ✅; 8 wala test: last 3 digits 200 → 100 → 50 → 25, 3 half poore ✅ → haan (7200 ÷ 24 = 300). 8136 ÷ 36: 36 = 4 × 9; 4 wala test: last 2 digits 36 → 18 → 9, 2 half poore ✅; 9 wala test: 8 + 1 + 3 + 6 = 18, 9 ki table mein ✅ → haan (8136 ÷ 36 = 226).
— 📌 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.
- 12345 ÷ 4 ka remainder = ?
- 12345 ÷ 5 ka remainder = ?
- 12345 ÷ 3 ka remainder = ?
- 12345 ÷ 10 ka remainder = ?
- 2026 ÷ 9 ka remainder = ?
- 2026 ÷ 4 ka remainder = ?
- 2026 ÷ 5 ka remainder = ?
- 777 ÷ 9 ka remainder = ?
- 777 ÷ 2 ka remainder = ?
- 10001 ÷ 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
- 1
- 0
- 0
- 5
- 1
- 2
- 1
- 3
- 1
- 2
Dhyaan (rule upar Type 4 ke box mein: ÷ 9 aur ÷ 3 ke liye digit sum ka bacha hua, ÷ 4 ke liye last 2 digits ka remainder): 12345 ÷ 3: digit sum 15 = 3 × 5 → remainder 0. 2026 ÷ 9: 2 + 0 + 2 + 6 = 10 → 1 + 0 = 1 → remainder 1. 2026 ÷ 4: last 2 digits 26 = 4 × 6 + 2 → remainder 2. 777 ÷ 9: 7 + 7 + 7 = 21 → 2 + 1 = 3 → remainder 3. 10001 ÷ 9: 1 + 0 + 0 + 0 + 1 = 2 → remainder 2.
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. 12 se divisible hone ke liye kaunse do tests?
Jawab aur reason dekho
Sahi jawab: (d) 3 aur 4
- (a) galat ❌: 2 aur 6 ka common divisor 2; 18 dono pass karta hai par 12 se nahi (18 = 12 × 1 + 6).
- (b) galat ❌: 2 aur 3 se sirf 6 ka pata chalta hai; 18 pass par 12 se nahi.
- (c) galat ❌: 1 se har number divisible; isse kuch nahi milta.
- (d) sahi ✅: 12 = 3 × 4, aur 3, 4 ka 1 ke alawa koi common divisor nahi.
Q2. 37□ ko 9 se divisible banana hai. □ = ?
Jawab aur reason dekho
Sahi jawab: (c) 8
- (a) galat ❌: 3 + 7 + 7 = 17 → nahi.
- (b) galat ❌: 3 + 7 + 0 = 10 → nahi.
- (c) sahi ✅: 3 + 7 + 8 = 18 → 378 ÷ 9 = 42.
- (d) galat ❌: 3 + 7 + 1 = 11 → nahi.
Q3. 1080, 18 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (b) Haan: even aur digit sum 9
- (a) galat ❌: 18 = 2 × 9 se rule ban jaata hai.
- (b) sahi ✅: 18 = 2 × 9: dono pass → 1080 ÷ 18 = 60.
- (c) galat ❌: Last digit 0 to 2 wala test pass karta hai.
- (d) galat ❌: 3 se divisible hona kaafi nahi (15 jaisa).
Q4. 12, 4 se aur 2 se divisible hai. 8 se?
Jawab aur reason dekho
Sahi jawab: (a) Nahi
- (a) sahi ✅: 12 = 8 × 1 + 4; 4 aur 2 ka common divisor 2, isliye pair nahi chalti.
- (b) galat ❌: 4 × 2 = 8 par pair galat hai; 12 fail karta hai.
- (c) galat ❌: Is number (12) ke liye pakka nahi.
- (d) galat ❌: 8 ka apna rule hai: 12 → 6 → 3: sirf 2 half, 3rd half nahi ho sakta → nahi.
Q5. 4□52 ko 3 se divisible: kitne digits □ mein aa sakte hain?
Jawab aur reason dekho
Sahi jawab: (d) 3 (1, 4, 7)
- (a) galat ❌: 1 ke alawa 4 aur 7 bhi chalte hain.
- (b) galat ❌: 7 bhi: 4 + 7 + 5 + 2 = 18.
- (c) galat ❌: Sirf woh digits jinse digit sum 3 ki table mein aaye.
- (d) sahi ✅: 11 + □ = 12, 15 ya 18 → □ = 1, 4, 7.
Q6. 12345 ÷ 9 ka remainder?
Jawab aur reason dekho
Sahi jawab: (c) 6
- (a) galat ❌: 5 sirf last digit hai.
- (b) galat ❌: 15 mein abhi ek 9 aur hai.
- (c) sahi ✅: Digit sum 15 → 15 − 9 = 6.
- (d) galat ❌: 0 tab hota jab digit sum 9 ki table mein hota.
Q7. 15 se divisible kaunsa hai?
Jawab aur reason dekho
Sahi jawab: (b) 2340
- (a) galat ❌: 5 ✅ par 2 + 3 + 3 + 5 = 13 (3 ❌).
- (b) sahi ✅: Last digit 0 (5 ✅), 2 + 3 + 4 + 0 = 9 (3 ✅) → 2340 ÷ 15 = 156.
- (c) galat ❌: 5 ✅ par 2 + 3 + 4 + 5 = 14 (3 ❌).
- (d) galat ❌: 5 ✅ par 2 + 3 + 5 + 0 = 10 (3 ❌).
Q8. 36, 24 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (a) Nahi
- (a) sahi ✅: 24 = 3 × 8; 8 wala test fail (36 → 18 → 9: 3rd half nahi ho sakta). 36 = 24 × 1 + 12.
- (b) galat ❌: 6 aur 4 ka common divisor 2, pair nahi chalti.
- (c) galat ❌: Digit sum 9 sirf 3 aur 9 ka test hai.
- (d) galat ❌: Even hona sirf 2 ka test hai.
Q9. Kaunsa number 2, 3, 4, 5 sab se divisible hai?
Jawab aur reason dekho
Sahi jawab: (d) 60
- (a) galat ❌: 30: 4 se nahi (30 = 4 × 7 + 2).
- (b) galat ❌: 40: 3 se nahi (4 + 0 = 4).
- (c) galat ❌: 45: odd, 2 aur 4 se nahi.
- (d) sahi ✅: 60: last digit 0 (2, 5 ✅), 6 + 0 = 6 (3 ✅), 60 = 4 × 15 (4 ✅).
Q10. 7392, 8 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (c) Haan, 392 → 196 → 98 → 49
- (a) galat ❌: Last digit akela 8 ke liye kuch nahi batata.
- (b) galat ❌: 11 se divisible hona 8 ke liye kuch nahi batata.
- (c) sahi ✅: Teen baar half poora → divisible (7392 ÷ 8 = 924).
- (d) galat ❌: Digit sum 8 ka test nahi hai.
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🧠 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 number kaunsa hai jo 2, 3, 4, 5 aur 6 sab se divisible ho?
Solution dekho
2 aur 5 ke liye last digit 0. 4 ke liye last 2 digits 4 se divisible: 20, 40, 60 … 3 ke liye digit sum 3 ki table mein: 20 (2) ❌, 40 (4) ❌, 60 (6) ✅. 6 = 2 aur 3, dono already ✅. Jawab 60.
Check: 60 ÷ 4 = 15, 60 ÷ 6 = 10 ✅
Q2. 738, 9 se divisible hai. Iske digits reverse karo: 837. Kya 837 bhi 9 se divisible hai? Kyun?
Solution dekho
Haan! Digit sum order se nahi badalta: 7 + 3 + 8 = 8 + 3 + 7 = 18. Isliye 837 ÷ 9 = 93 ✅. (Par 4 ya 8 ke liye aisa nahi: 116 ÷ 4 = 29, par 611 odd hai, 4 se nahi.)
Q3. Teen consecutive numbers ka multiply (jaise 7 × 8 × 9) hamesha 6 se divisible kyun hota hai? (Consecutive numbers = ek ke baad ek aane wale numbers, jaise 7, 8, 9.)
Solution dekho
Teen consecutive numbers mein kam se kam ek even hota hai (2 wala test ✅), aur ek zaroor 3 ki table mein hota hai (3, 6, 9 … har teesra number). To multiply 2 se bhi aur 3 se bhi divisible → 6 se divisible. 7 × 8 × 9 = 504, aur 504 ÷ 6 = 84 ✅.
Q4. Ek 4-digit number 8□□0 hai jo 11 aur 9 dono se divisible hai (□□ do digits, same ya alag). Ek aisa number dhoondo.
Solution dekho
11 ke liye: right se 1st + 3rd = 0 + □(hundreds), 2nd + 4th = □(tens) + 8. 9 ke liye: 8 + □ + □ + 0, 9 ki table mein. Try 8910: 11 test: (0 + 9) aur (1 + 8) → difference 0 ✅; 9 test: 8 + 9 + 1 + 0 = 18 ✅. Jawab 8910.
Check: 8910 ÷ 99 = 90 (99 = 9 × 11) ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Missing digit, “kis-kis se divisible”, combined rules. |
| Maths Olympiad | Sabse chhota/bada number, consecutive numbers (jaise 7, 8, 9), digits reverse karna. |
| NTSE / MAT | Number puzzles aur tez Simplification. |
| SSC / Bank (aage) | Number system: divisibility aur remainder ke tricky sawaal. |
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. 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: “Divisor identify karo, bada ho to break karo, dono test pass.” Chapter 17 poora ho gaya! 🎉 Ab Chapter 18 mein digit sum ka ek aur kamaal: 9 wala check, jisse addition, subtraction aur multiplication ka answer 5 second mein check ho jaata hai! 🧩
— 📌 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)
12 = 3 × 4. 3 wala test: 4 + 3 + 5 + 6 = 18 ✅. 4 wala test: last 2 digits 56 → 28 → 14 ✅. Dono pass → 12 se divisible ✅.
| 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. 12 ke liye 2 aur 6 kyun nahi?
2 aur 6 ka common divisor 2 hai. 18 dono pass karta hai par 12 se divisible nahi. 3 aur 4 lo.
2. 24 ke liye kaunsi pair?
3 aur 8 (sirf 1 common). 4 aur 6 nahi chalegi.
3. Missing digit mein kai jawab ho sakte hain?
Haan, aksar. 4□52 ÷ 3 mein teen: 1, 4, 7. Sab likho.
4. 7 ka rule kyun nahi seekha?
7 ka koi aasaan, bharose wala shortcut nahi; seedha divide karna jaldi hai.
5. Agla chapter kya hai?
Chapter 18: Digit-Sum Checking, yaani 9 wala check: digit sum se answer check karna. Jald aa raha hai.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Pehle divisor identify karo: last digit, last 2/3 digits, digit sum ya alternate sum.
- Bada divisor: do aise numbers mein break karo jinka 1 ke alawa koi common divisor nahi (12 = 3 × 4).
- Dono tests pass → divisible; ek bhi fail → nahi.
- Missing digit: rule reverse chalao, saare jawab likho.
- Dhokhe: 4 aur 2 se 8 nahi banta; 6 aur 4 se 24 nahi banta.
— charumam.com 🎯
👉 Aage Kya?
Chapter 17 poora! 🎉 Agla Chapter 18: Digit-Sum Checking (digit sum se addition, subtraction aur multiplication ke answers 5 second mein check karna) jald aa raha hai.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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