Chapter 17 shuru! 🎉 Chapter 16 mein divide karna seekha. Ab ek aur tez sawaal: bina divide kiye kaise pata karein ki remainder 0 aayega ya nahi? Isi ko divisibility kehte hain. Aaj pehle teen rules: 2, 5 aur 10. Teeno mein bas last digit dekhna hai! 4738 ko 2 se divide karo to kuch bachega? Last digit 8 even hai, to nahi. Ek second mein. Saath mein “divisible”, “last digit”, even aur odd ka matlab bilkul shuru se.
⏱ Before Timer: Pehle Purane Method Se
Pehle school wala “long division” (ek doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). 572 ÷ 2:
- Left se pehla digit 5. 5 mein 2 kitni baar? 2 baar (2 × 2 = 4). Upar 2 likho. 5 − 4 = 1 bacha.
- Agla digit 7 neeche utaaro, bache 1 ke saath: 17. 17 mein 2 kitni baar? 8 baar (2 × 8 = 16). Upar 8 likho. 17 − 16 = 1 bacha.
- Agla digit 2 neeche utaaro: 12. 12 mein 2 kitni baar? 6 baar (2 × 6 = 12). Upar 6 likho. 12 − 12 = 0 bacha.
- Quotient 286, remainder 0. Remainder 0, to 572, 2 se divisible hai (check: 2 × 286 = 572).
Har step pe “kitni baar”, multiply, subtract. 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 2 se divide kijiye aur dekhiye remainder kitna aata hai, copy-pencil ke saath. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
Haan. 4738 ÷ 2 = 2369, remainder 0, to divisible.
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 16.1 ka ÷, quotient aur remainder; Chapter 1 ka place value (units, tens). Naya: divisible, last digit, even/odd. Sab neeche ke boxes mein. Naye word:
📖 Word jo is post mein aayenge
- Divisor: jis number se divide karte hain: 12 ÷ 4 mein 4.
- Divisible: divide karne pe remainder 0. 24, 4 se divisible hai: 24 ÷ 4 = 6, kuch nahi bacha.
- Divisibility rule (test): bina poora division kiye remainder 0 hai ya nahi, yeh batane wala shortcut.
- Last digit: sabse right wala digit, yaani units ka digit. 3746 ka last digit 6.
- Even number: jo 2-2 ki pairs mein poora divide ho jaaye: 0, 2, 4, 6, 8, 10 …
- Odd number: jisme pairs ke baad ek akela bache: 1, 3, 5, 7, 9, 11 …
- Table (multiplication table): 5 ki table: 5, 10, 15, 20 … Jo number kisi ki table mein aaye, woh us number se divisible.
📏 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.
🔢 “Last digit”, “last 2 digits”, “last 3 digits” ka matlab
Number ko left se right padhte hain, to last digit matlab sabse right wala digit, yaani units wala digit (Chapter 1 ka place value). 3746 ka last digit 6.
Last 2 digits matlab sabse right ke do digits, ek saath ek 2-digit number ki tarah padho: 3716 ke last 2 digits 16 (solah), yaani tens ka 1 aur units ka 6. Last 3 digits: sabse right ke teen digits ek saath: 53128 ke last 3 digits 128.
Number ko do parts mein break karna (place value se): 3746 = 3740 + 6 (last digit alag). 3716 = 3700 + 16 (last 2 digits alag). 53128 = 53000 + 128 (last 3 digits alag). Left wale part ke peeche hamesha 0, 00 ya 000 hote hain.
Zero wale cases: 3704 ke last 2 digits “04”, matlab bas 4. 5300 ke last 2 digits “00”, matlab 0.
⚖️ Even aur odd numbers
Even number = jo 2-2 ki pairs mein poora divide ho jaaye, koi akela na bache: 2, 4, 6, 8, 10, 12 … Odd number = jisme pairs banane ke baad ek akela bach jaaye: 1, 3, 5, 7, 9, 11 …
Even numbers ka last digit hamesha 0, 2, 4, 6 ya 8 hota hai. Odd numbers ka last digit hamesha 1, 3, 5, 7 ya 9. (0 bhi even hai: 0 cheezon mein koi akela nahi bachta.)
🔁 Chhota sa warm-up
- 30 ÷ 2 = ?
- 35 ÷ 5 = ?
- 23 ÷ 5 → quotient aur remainder?
Answers dekho
- 15
- 7
- 4, remainder 3 (5 × 4 = 20, 20 + 3 = 23)
Pehle do mein remainder 0: 30, 2 se divisible; 35, 5 se divisible. Teesre mein remainder 3: 23, 5 se divisible nahi. Aaj yahi bina divide kiye pata karenge!
— 📌 charumam.com
🎬 Story: Sports Day Ki Pairs
Sports day pe 346 students aaye. Teacher ne kaha: “Sab 2-2 ki pairs mein khade ho jao. Koi akela to nahi bachega?”
Monitor Aarav count shuru karne wala tha. Zoya boli: “Ruko! 346 ka last digit 6 hai, aur 6 even hai. To koi akela nahi bachega!”
Pairs bani: 173 pairs, ek bhi student akela nahi. 🏃 “Bas last digit dekh kar?” Zoya: “Haan, kyunki har 10 students to pairs mein aaram se khade ho jaate hain!” Aaj yahi trick.
— 📌 charumam.com
🪄 Trick Step by Step: Sirf Last Digit Dekho
💡 Is post ki ek badi baat: Kisi number ka 2, 5 ya 10 se divisible hona sirf uske last digit pe depend karta hai. 2: last digit 0, 2, 4, 6 ya 8 (even). 5: last digit 0 ya 5. 10: last digit 0. Kyunki baaki number (tens se left) hamesha 10 ke groups hain, aur 10 = 2 × 5, jo 2, 5 aur 10 teeno se divisible hai.
Step 1: Last digit dhoondo
Number ka sabse right wala digit dekho. 3746 → last digit 6. 1235 → 5. 4590 → 0.
Step 2: 2 ka rule
Last digit 0, 2, 4, 6 ya 8 (even) → 2 se divisible. 3746: last digit 6, even → divisible. 1235: last digit 5, odd → nahi.
Step 3: 5 ka rule
Last digit 0 ya 5 → 5 se divisible. 1235: 5 → divisible. 4590: 0 → divisible. 3746: 6 → nahi.
Step 4: 10 ka rule
Last digit 0 → 10 se divisible. 4590 → divisible. 1235 → nahi (last digit 5, 0 nahi).
Step 5: Check karo (asli division se)
Divisor × quotient + remainder = number? 3746 ÷ 2 = 1873, kyunki 2 × 1873 = 3746 ✅. 1235 ÷ 2 → quotient 617, remainder 1: 2 × 617 = 1234, aur 1234 + 1 = 1235 ✅ (remainder 1, isliye divisible nahi).
Aur examples, ek ke baad ek
5678 ÷ 2? Last digit (units) 8: 8 even hai (0, 2, 4, 6, 8 mein se) → ✅ divisible. Check: 5678 ÷ 2 = 2839.
5678 ÷ 5? Last digit 8: 0 ya 5 nahi hai → ❌ divisible nahi. Check: 5678 = 5 × 1135 + 3.
5678 ÷ 10? Last digit 8: 0 nahi hai → ❌ divisible nahi. Check: 5678 = 10 × 567 + 8.
905 ÷ 2? Last digit (units) 5: 5 odd hai (1, 3, 5, 7, 9 mein se) → ❌ divisible nahi. Check: 905 = 2 × 452 + 1.
905 ÷ 5? Last digit 5: 0 ya 5 hai → ✅ divisible. Check: 905 ÷ 5 = 181.
905 ÷ 10? Last digit 5: 0 nahi hai → ❌ divisible nahi. Check: 905 = 10 × 90 + 5.
Bonus: remainder bhi turant
÷ 2: odd number ka remainder hamesha 1 (ek akela bachta hai). ÷ 10: remainder = last digit khud (5678 ÷ 10 → 567, remainder 8). ÷ 5: last digit 0 se 4 ho to wahi remainder; 5 se 9 ho to last digit − 5 (Post 16.4): 5678 ka last digit 8 → 8 − 5 = 3. Check: 5678 = 5 × 1135 + 3 ✅.
🆘 Atak gaye? Aise socho
- Divisible = remainder 0. Kuch bacha to divisible nahi.
- Sirf last digit dekho, pehla digit nahi.
- 2: 0, 2, 4, 6, 8. 5: 0, 5. 10: sirf 0.
- Shaq ho to Chapter 16 ki trick se divide karke check karo.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Har number = 10 ke groups + last digit. 3746 = 3740 + 6. 3740 matlab 374 tens, yaani 10-10 ke 374 groups. Har 10 ko 2 se divide karo to 5 (koi remainder nahi), 5 se divide karo to 2, 10 se divide karo to 1. Matlab 10 ke groups hamesha 2, 5 aur 10 se divisible. Diagram dekho.
To decision sirf bache hue last digit se hota hai: 6 ÷ 2 = 3, remainder 0, to 3746 bhi 2 se divisible. 6 ÷ 5 → remainder 1, to 3746 bhi 5 se divisible nahi (remainder 1).
3 ka aisa rule kyun nahi?
10, 3 se divisible nahi: 10 = 3 × 3 + 1 (1 bachta hai). To har 10 ke group se 1 bachta jaata hai, aur last digit akela decide nahi kar sakta. Jaise 13: last digit 3, par 13 = 3 × 4 + 1, divisible nahi! Isliye 3 ka rule alag hai (agla post: digit sum).
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: 5670: 2, 5 aur 10 se divisible?
5670 ÷ 2? Last digit (units) 0: 0 even hai (0, 2, 4, 6, 8 mein se) → ✅ divisible. Check: 5670 ÷ 2 = 2835.
5670 ÷ 5? Last digit 0: 0 ya 5 hai → ✅ divisible. Check: 5670 ÷ 5 = 1134.
5670 ÷ 10? Last digit 0: 0 hai → ✅ divisible. Check: 5670 ÷ 10 = 567.
Teeno se divisible. (Last digit 0 wala number hamesha teeno se divisible.)
✔️ Normal method se check: 567 × 10 = 5670 ✅
Example 2: 2345: 2, 5 aur 10 se divisible?
2345 ÷ 2? Last digit (units) 5: 5 odd hai (1, 3, 5, 7, 9 mein se) → ❌ divisible nahi. Check: 2345 = 2 × 1172 + 1.
2345 ÷ 5? Last digit 5: 0 ya 5 hai → ✅ divisible. Check: 2345 ÷ 5 = 469.
2345 ÷ 10? Last digit 5: 0 nahi hai → ❌ divisible nahi. Check: 2345 = 10 × 234 + 5.
Sirf 5 se divisible.
✔️ Normal method se check: 469 × 5: 4690 ka half 2345 (Chapter 5) ✅
Example 3: ₹ 785 ko sirf ₹ 5 ke coins mein exact badal sakte hain (kuch na bache)?
785 ÷ 5? Last digit 5: 0 ya 5 hai → ✅ divisible. Check: 785 ÷ 5 = 157.
Haan. Kitne coins? Post 16.1: double 1570, 0 remove karo → 157 coins.
✔️ Normal method se check: 157 × 5: 1570 ka half 785 ✅
Example 4: 98 students 2-2 ki pairs mein: koi akela bachega?
98 ÷ 2? Last digit (units) 8: 8 even hai (0, 2, 4, 6, 8 mein se) → ✅ divisible. Check: 98 ÷ 2 = 49.
Nahi bachega: 49 pairs.
✔️ Normal method se check: 49 × 2 = 98 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
CharuMam ki smart trick: “Saara number mat dekho, sirf woh part dekho jo decision karta hai.” 10 ke saare groups pehle se 2, 5 aur 10 se divisible hain, isliye last digit hi boss hai. Yahi soch agle posts mein bhi: 4 ke liye last 2 digits, 8 ke liye last 3 digits, aur 3 aur 9 ke liye digit sum.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| Remainder 0 hai ya nahi, bas itna jaanna (÷ 2, 5, 10) | ✅ Haan | Last digit se 1 second mein. |
| Pairs, ₹ 5 ke coins, ₹ 10 ke notes | ✅ Haan | Exact banega ya kuch bachega. |
| Quotient bhi chahiye (5670 ÷ 5 kitna?) | ⚠️ Rule + Chapter 16 | Rule sirf haan/nahi batata; quotient ke liye Post 16.1. |
| ÷ 3, ÷ 4, ÷ 9 jaise | ❌ Is rule se nahi | Inke apne rules hain (agle posts). |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Pehla digit dekh liya
Sawaal: 47, 2 se divisible?
❌ Galat: “Pehla digit 4 even hai, to haan.”
✅ Sahi: last digit dekho: 7 odd → nahi. 47 = 2 × 23 + 1.
Mistake 2: 5 ke rule mein 0 bhool gaye
Sawaal: 340, 5 se divisible?
❌ Galat: “Last digit 5 nahi, to nahi.”
✅ Sahi: 0 bhi chalta hai: 340 ÷ 5 = 68 → divisible.
Mistake 3: Quotient dekh kar decision
Sawaal: 12, 5 se divisible?
❌ Galat: “12 ÷ 5 → 2 aaya, jawab aa gaya to divisible.”
✅ Sahi: remainder dekho: 12 = 5 × 2 + 2, remainder 2 → nahi. Divisible sirf tab jab remainder 0.
Mistake 4: 5 se divisible to 10 se bhi?
Sawaal: 15, 10 se divisible?
❌ Galat: “5 se divisible hai, to 10 se bhi.”
✅ Sahi: 10 ke liye last digit 0 chahiye. 15 = 10 × 1 + 5 → nahi. (Reverse sahi hai: 10 se divisible number 5 aur 2 se bhi divisible.)
🧠 Rule of thumb: Remainder 0 = divisible. 2: last digit even (0, 2, 4, 6, 8). 5: last digit 0 ya 5. 10: last digit 0.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: har ek mein “Haan” ya “Nahi”, aur kuch remainder wale. Target: 2 minute.
- 346: 2 se divisible?
- 1235: 2 se divisible?
- 4590: 10 se divisible?
- 785: 5 se divisible?
- 902: 5 se divisible?
- 1001: 2 se divisible?
- 7770: 10 se divisible?
- 6005: 10 se divisible?
- 88: 2 se divisible?
- 4445: 5 se divisible?
- 3330: 5 se divisible?
- 1112: 10 se divisible?
- 57 ÷ 2 ka remainder = ?
- 5678 ÷ 10 ka remainder = ?
- 143 ÷ 5 ka remainder = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- Haan (last digit 6 even)
- Nahi (last digit 5 odd)
- Haan (last digit 0)
- Haan (last digit 5)
- Nahi (last digit 2)
- Nahi (last digit 1 odd)
- Haan (last digit 0)
- Nahi (last digit 5)
- Haan (last digit 8 even)
- Haan (last digit 5)
- Haan (last digit 0)
- Nahi (last digit 2)
- 1
- 8
- 3
Kaise kiya (kuch): 6005: last digit 5, 0 nahi → 10 se nahi (par 5 se haan!). 143 ÷ 5: last digit 3 (0 se 4 ke beech) → remainder 3. 57 ÷ 2: odd → remainder 1.
— 📌 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.
- 48 ÷ 2 ka remainder = ?
- 91 ÷ 2 ka remainder = ?
- 250 ÷ 5 ka remainder = ?
- 254 ÷ 5 ka remainder = ?
- 258 ÷ 5 ka remainder = ?
- 370 ÷ 10 ka remainder = ?
- 375 ÷ 10 ka remainder = ?
- 1003 ÷ 2 ka remainder = ?
- 999 ÷ 5 ka remainder = ?
- 4321 ÷ 10 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
- 4
- 3
- 0
- 5
- 1
- 4
- 1
Dhyaan: 258 ÷ 5: last digit 8 → 8 − 5 = 3. 999 ÷ 5: last digit 9 → 9 − 5 = 4. 4321 ÷ 10: remainder = last digit 1.
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. 3746, 2 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (a) Haan, last digit 6 even hai
- (a) sahi ✅: Last digit 6 even → divisible (3746 ÷ 2 = 1873).
- (b) galat ❌: Pehla digit nahi, last digit dekhna hota hai.
- (c) galat ❌: Beech ka digit 7 kuch decide nahi karta; sirf last digit.
- (d) galat ❌: Shortcut se 1 second mein pata chalta hai, poora division zaroori nahi.
Q2. Kaunsa number 5 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (d) 4590
- (a) galat ❌: Last digit 2 → remainder 2.
- (b) galat ❌: Last digit 3 → remainder 3.
- (c) galat ❌: Last digit 8 → remainder 8 − 5 = 3.
- (d) sahi ✅: Last digit 0 → 5 se divisible (4590 ÷ 5 = 918).
Q3. “Divisible” ka matlab kya hai?
Jawab aur reason dekho
Sahi jawab: (c) Divide karne pe remainder 0
- (a) galat ❌: Bada ya chhota hona kuch decide nahi karta.
- (b) galat ❌: Kuch bache to divisible nahi.
- (c) sahi ✅: Kuch nahi bachta: remainder 0.
- (d) galat ❌: Quotient 0 ka matlab number divisor se chhota hai, divisible nahi.
Q4. 7770 kis-kis se divisible hai?
Jawab aur reason dekho
Sahi jawab: (b) 2, 5 aur 10 teeno
- (a) galat ❌: Last digit 0 hai, teeno se divisible.
- (b) sahi ✅: Last digit 0 → 2, 5, 10 teeno.
- (c) galat ❌: Last digit 0 to 2 aur 5 se bhi divisible.
- (d) galat ❌: 10 bhi: last digit 0 hai.
Q5. 6005, 10 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (a) Nahi, last digit 5 hai
- (a) sahi ✅: 10 ke liye last digit 0 chahiye; yahan 5. Remainder 5.
- (b) galat ❌: Beech ke 0 kuch nahi badalte; sirf last digit.
- (c) galat ❌: 5 se divisible hona kaafi nahi; 10 ke liye 0 chahiye.
- (d) galat ❌: Pehla digit nahi, last digit dekhte hain.
Q6. 57 ÷ 2 ka remainder?
Jawab aur reason dekho
Sahi jawab: (d) 1
- (a) galat ❌: 0 tab hota jab last digit even hota.
- (b) galat ❌: 7 last digit hai, remainder nahi; ÷ 2 mein remainder sirf 0 ya 1.
- (c) galat ❌: 28 quotient hai, remainder nahi.
- (d) sahi ✅: 57 odd → ek akela bachta hai: remainder 1 (57 = 2 × 28 + 1).
Q7. 2-2 ki pairs mein koi akela nahi bachega: kitne students?
Jawab aur reason dekho
Sahi jawab: (c) 418
- (a) galat ❌: Last digit 1 odd → ek akela.
- (b) galat ❌: Last digit 5 odd → ek akela.
- (c) sahi ✅: 418 even (last digit 8) → 209 pairs.
- (d) galat ❌: Last digit 9 odd → ek akela.
Q8. 143 ÷ 5 ka remainder?
Jawab aur reason dekho
Sahi jawab: (b) 3
- (a) galat ❌: 8 kahin nahi aata; last digit 3 hai.
- (b) sahi ✅: Last digit 3 (0 se 4 ke beech) → remainder 3. Check: 5 × 28 + 3 = 143.
- (c) galat ❌: 0 tab jab last digit 0 ya 5 ho.
- (d) galat ❌: Pehla digit 1 remainder nahi hota.
Q9. 2, 5 aur 10 ke rules last digit se kyun chalte hain?
Jawab aur reason dekho
Sahi jawab: (a) 10 khud 2, 5 aur 10 se divisible hai (10 = 2 × 5)
- (a) sahi ✅: Baaki number 10 ke groups hain, jo hamesha divisible; sirf last digit bachta hai.
- (b) galat ❌: Last digit chhota-bada kuch bhi ho sakta hai; yeh reason nahi.
- (c) galat ❌: Har number even nahi hota (7, 13 odd hain).
- (d) galat ❌: 10 even hai, odd nahi.
Q10. Kaunsa statement sahi hai?
Jawab aur reason dekho
Sahi jawab: (d) 10 se divisible number, 5 se bhi divisible hota hai
- (a) galat ❌: 15, 5 se divisible hai par 10 se nahi.
- (b) galat ❌: 12, 2 se divisible hai par 10 se nahi.
- (c) galat ❌: 15 odd hai aur 5 se divisible hai.
- (d) sahi ✅: Last digit 0 → 5 ka rule bhi pass (0 ya 5).
— 📌 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. Ek number 2 se bhi divisible hai aur 5 se bhi. Uska last digit kya hoga?
Solution dekho
2 ke liye last digit 0, 2, 4, 6, 8. 5 ke liye 0 ya 5. Dono lists mein common sirf 0. To last digit 0, aur aisa number 10 se bhi divisible hota hai (jaise 70: 70 ÷ 2 = 35, 70 ÷ 5 = 14, 70 ÷ 10 = 7).
Q2. 47□ (□ = ek digit) ko 5 se divisible banana hai. □ kya-kya ho sakta hai? Aur 2 aur 5 dono se?
Solution dekho
5 ke liye last digit 0 ya 5: 470 ya 475. 2 aur 5 dono ke liye sirf 0: 470.
Check: 475 ÷ 5 = 95, aur 470 ÷ 10 = 47 ✅
Q3. 1 se 100 tak kitne numbers 5 se divisible hain? Aur kitne 10 se?
Solution dekho
5 se: 5, 10, 15 … 100, yaani 5 ki table ke pehle 20 numbers (5 × 20 = 100): 20. 10 se: 10, 20 … 100: 10. (Har doosra 5 wala number 10 se bhi divisible hai.)
Q4. Do odd numbers add karo to jawab even hoga ya odd? 37 + 45 se socho.
Solution dekho
37 + 45 = 82, last digit 2 → even. Kyun: dono mein ek-ek akela bachta hai (pairs ke baad), aur do akele mil kar ek pair ban jaate hain. Isliye odd + odd = hamesha even.
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🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Divisibility rules, even/odd, “kaunsa number 5 se divisible hai” jaise sawaal. |
| Maths Olympiad | Missing digit (47□), counting (1 se 100 mein kitne). |
| NTSE / MAT | Number series aur tez Simplification mein remainder dekhna. |
| SSC / Bank (aage) | Number system ke sawaal: divisibility pe hi based. |
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.
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👩🏫 CharuMam Ka Tip
Mantra: “Last digit hi boss.” Bus number, house number, bill: kahin bhi number dikhe to socho: 2 se? 5 se? 10 se? Daily 10 numbers. Agle post mein 3 aur 9: wahan last digit nahi, saare digits ka sum dekhna hai! 🔎
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⏱ 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)
Last digit 8. 8 even hai (0, 2, 4, 6, 8 mein se), to divisible ✅. Poora division karne ki zaroorat hi nahi!
| 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. 🚀
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🤖 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:
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❓ FAQs (Students Ke Asli Sawaal)
1. Divisible aur “poora divide” mein difference?
Koi difference nahi. Dono ka matlab: remainder 0.
2. 0, 2 se divisible hai?
Haan: 0 ÷ 2 = 0, remainder 0. Isliye 0 even hai, aur 10, 20, 30 jaise numbers 2 se divisible.
3. Bade number (jaise 98765430) pe bhi rule chalega?
Haan, kitna bhi bada ho: last digit 0, to 2, 5, 10 teeno se divisible.
4. Rule se quotient bhi milega?
Nahi, rule sirf haan/nahi batata. Quotient ke liye Chapter 16 ki tricks.
5. Agle post mein kya hai?
Post 17.2: 3 aur 9 ke rules (digit sum se) aur 6 ka rule (2 aur 3 dono).
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- Divisible = divide karne pe remainder 0.
- Last digit = sabse right wala digit (units).
- 2: last digit 0, 2, 4, 6, 8 (even). 5: 0 ya 5. 10: sirf 0.
- Kyun: baaki number 10 ke groups hain, aur 10 = 2 × 5.
- Check hamesha asli division se: divisor × quotient + remainder = number.
— charumam.com 🎯
👉 Aage Kya?
Ab Post 17.2: 3 aur 9 ke rules. Yahan last digit kaam nahi karta (13 ka last digit 3, par 13, 3 se divisible nahi!). Secret hai digit sum: 234 → 2 + 3 + 4 = 9, to 234, 9 se divisible!
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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