Post 17.1 mein last digit, 17.2 mein digit sum. Aaj teen naye rules: 4 ke liye sirf last 2 digits dekho, 8 ke liye last 3 digits. Kyunki 100 = 4 × 25 aur 1000 = 8 × 125. Aur 11 ka rule sabse mazedaar: right se digits pe baari-baari + aur −, yaani alternate digits ka difference! 7392 → (2 + 3) aur (9 + 7) ka difference 11 → divisible. Har rule ka check asli division 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). 916 ÷ 4:
- 9 mein 4 kitni baar? 2 baar (4 × 2 = 8). Upar 2. Bacha 9 − 8 = 1.
- Agla digit 1 neeche, bache 1 ke saath: 11. 11 mein 4: 2 baar (8). Upar 2. Bacha 3.
- Agla digit 6 neeche: 36. 36 mein 4: 9 baar (36). Upar 9. Bacha 0.
- Quotient 229, remainder 0 → 916, 4 se divisible (check: 4 × 229 = 916).
Har step pe table, subtract, digit neeche. 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 8 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. 53128 ÷ 8 = 6641, 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” aur last digits; Chapter 5 ka half; Chapter 4 ki 11 ki table. Naya: last 2/3 digits wala test aur alternate digits. Sab neeche ke boxes mein. Naye word:
📖 Word jo is post mein aayenge
- Last 2 digits: sabse right ke do digits, ek 2-digit number ki tarah: 3716 → 16.
- Last 3 digits: sabse right ke teen digits: 53128 → 128.
- Half: number ke do equal parts mein se ek: 28 ka half 14.
- Whole number: 0, 1, 2, 3 … jaise number, jismein half ya piece na ho: 7 whole number hai, 4 aur half nahi.
- Alternate (baari-baari): ek chhod kar ek: right se 1st, 3rd, 5th … ek group; 2nd, 4th … doosra group.
- Difference: bade number mein se chhota subtract karna: 16 aur 5 ka difference 16 − 5 = 11.
- Leap year: woh year jisme February 29 days ka hota hai; aam taur pe year ka number 4 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.
✅ “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.
✂️ Half karna aur “4 = 2 × 2, 8 = 2 × 2 × 2”
Chapter 5 mein half kiya tha: 28 ka half 14 (28 = 14 + 14). 4 se divide = do baar half (kyunki 4 = 2 × 2): 28 → 14 → 7, to 28 ÷ 4 = 7. 8 se divide = teen baar half (8 = 2 × 2 × 2): 128 → 64 → 32 → 16, to 128 ÷ 8 = 16.
Odd kab problem hai? Sirf tab, jab odd number aaye aur abhi half karna baaki ho. Jaise 4 ke liye 2 half chahiye: 18 → 9 (sirf 1 half hua). 9 odd hai aur 2nd half abhi baaki hai, par 9 ka half whole number nahi (4 aur half). Matlab 18, 4 se divisible nahi. Agar odd number last zaroori half ke baad aaye, to koi problem nahi: 28 → 14 → 7, do half poore, 7 odd hai par ab half baaki nahi, to 28, 4 se divisible (28 = 4 × 7).
÷ 8 ke liye dono saath-saath: 136 → 68 → 34 → 17: teeno half poore ho gaye, 17 odd hai par ab koi half baaki nahi, to haan (136 = 8 × 17). | 124 → 62 → 31: sirf 2 half hue, 31 odd hai aur 3rd half nahi ho sakta, to nahi (124 = 8 × 15 + 4).
🔁 Chapter 4 yaad karo: 11 ki table
11 ki table: 11, 22, 33, 44, 55, 66, 77, 88, 99. Aur 11 × 9 = 99 (Chapter 4: 9 × 11 → 9 | 9). 99 aaj bahut kaam aayega: 100 = 99 + 1.
🔁 Chhota sa warm-up
- 28 ka half, phir uska half = ?
- 100 ÷ 4 = ?
- 11 × 9 = ?
Answers dekho
- 7 (28 → 14 → 7)
- 25
- 99
28 → 14 → 7: do baar half, yaani 28 ÷ 4 = 7. Aur 100 = 4 × 25: isliye hundreds wala part hamesha 4 se divisible. Aaj yahi do baatein rule banengi!
— 📌 charumam.com
🎬 Story: Dadi Ka Birthday
Riya ki Dadi ka birthday 29 February ko hai, jo sirf leap year mein aata hai. Riya ne poocha: “Dadi ka agla asli birthday kab, 2028 mein?”
Papa bole: “Leap year ka number aam taur pe 4 se divisible hota hai. 2028 ÷ 4 karo.” Riya ne kaha: “Poora divide kyun? Sirf last 2 digits: 28, aur 28 = 4 × 7. To 2028 leap year hai!”
Dadi muskurayi: “Shabaash! 🎂 Par last 2 digits hi kyun?” Riya: “Kyunki 100 = 4 × 25, to 2000 to hamesha 4 se divisible hai!” Aaj yahi trick, aur 8 aur 11 ki bhi.
— 📌 charumam.com
🪄 Trick Step by Step: 4: Last 2 Digits, 8: Last 3 Digits, 11: Alternate Sum
💡 Is post ki ek badi baat: 4 ka rule: last 2 digits wala number 4 se divisible (00 bhi chalega) → full number bhi 4 se divisible (jaise 3716: last 2 digits 16 = 4 × 4, to poora 3716 bhi 4 se divisible). 8 ka rule: last 3 digits wala number 8 se divisible → full number bhi 8 se divisible (jaise 53128: last 3 digits 128 = 8 × 16, to poora 53128 bhi 8 se divisible). Test karne ke liye: 4 ke liye 2 baar half, 8 ke liye 3 baar half. Agar half karna baaki ho aur beech mein odd number aa jaaye, to nahi. Saare zaroori half poore ho gaye, to haan, chahe last number odd ho (136 → 68 → 34 → 17: haan). 11 ka rule: right se 1st, 3rd, 5th … digits ka sum, aur 2nd, 4th … ka sum. In dono ka difference 0, 11, 22 … ho → 11 se divisible.
Step 1: 4: last 2 digits alag karo
3716 → last 2 digits 16. Baaki 3700 to hamesha 4 se divisible (100 = 4 × 25).
Step 2: Last 2 digits ko test karo
16 ko 2 baar half: 16 → 8 → 4, dono baar whole number → 4 se divisible. To 3716 bhi. (16 = 4 × 4.)
Step 3: 8: last 3 digits
53128 → last 3 digits 128 (baaki 53000, kyunki 1000 = 8 × 125). 128 ko 3 baar half: 128 → 64 → 32 → 16 → 8 se divisible.
Step 4: 11: alternate digits
7392: right se places count karo. 1st (2) aur 3rd (3): 2 + 3 = 5. 2nd (9) aur 4th (7): 9 + 7 = 16. Difference: 16 − 5 = 11 → 11 ki table mein → 11 se divisible.
Step 5: Check karo (asli division se)
3716 ÷ 4 = 929, kyunki 4 × 929 = 3716 ✅. 53128 ÷ 8 = 6641 ✅. 7392 ÷ 11 = 672, kyunki 11 × 672 = 7392 ✅.
Aur examples, ek ke baad ek
5318 ÷ 4? Last 2 digits: 18. 18 ÷ 4: 18 = 4 × 4 + 2 → ❌ divisible nahi. Check: 5318 = 4 × 1329 + 2.
18 ko 2 baar half karo: 18 → 9: 9 odd hai aur abhi half baaki tha, iska half whole number nahi ❌.
7100 ÷ 8? Last 3 digits: 100. 100 ÷ 8: 100 = 8 × 12 + 4 → ❌ divisible nahi. Check: 7100 = 8 × 887 + 4.
100 ko 3 baar half karo: 100 → 50 → 25: 25 odd hai aur abhi half baaki tha, iska half whole number nahi ❌.
4825 ÷ 11? Right se digits: 1st, 3rd, 5th … place wale: 5 + 8 = 13. 2nd, 4th … place wale: 2 + 4 = 6. Difference (bada − chhota): 13 − 6 = 7. 7 0 ya 11, 22 … nahi hai → ❌ divisible nahi. Check: 4825 = 11 × 438 + 7.
1342 ÷ 11? Right se digits: 1st, 3rd, 5th … place wale: 2 + 3 = 5. 2nd, 4th … place wale: 4 + 1 = 5. Difference (bada − chhota): dono 5, difference 0. 0 0 hai → ✅ divisible. Check: 1342 ÷ 11 = 122.
Bonus: ÷ 4 ka remainder bhi last 2 digits se
5318 ÷ 4 ka remainder = 18 ÷ 4 ka remainder = 2 (18 = 4 × 4 + 2). Check: 5318 = 4 × 1329 + 2 ✅. (8 ke liye bhi yahi: last 3 digits ka remainder.)
🆘 Atak gaye? Aise socho
- 4: sirf last 2 digits; 2 baar half, dono baar whole number?
- 8: sirf last 3 digits; 3 baar half.
- 11: right se 1st, 3rd … ek group; 2nd, 4th … doosra. Sum alag-alag, phir difference.
- Difference 0, 11, 22, 33 … ho to divisible.
- Shaq ho to asli division se check.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
4 aur 8: hundreds aur thousands hamesha divisible. 3716 = 3700 + 16. 3700 = 37 hundreds, aur har 100 = 4 × 25, yaani 4 se divisible. Isliye sirf 16 decide karta hai. Isi tarah 1000 = 8 × 125, to 8 ke liye thousands wala part hamesha divisible; sirf last 3 digits bachte hain. Diagram dekho.
11 kyun alternate? Step by step
- Har ten, 11 se 1 kam hai: 10 = 11 − 1.
- Chhota example, 77: 77 = 70 + 7 (7 tens aur 7 units).
- Tens wala part: 70 = 7 × 10. Har 10 = 11 − 1, to 7 tens = 7 × 11 − 7. (Check: 7 × 11 = 77, aur 77 − 7 = 70 ✅)
- Ab add karo: 77 = (7 × 11 − 7) + 7. Tens digit wala −7 aur units digit wala +7 cancel ho gaye (−7 + 7 = 0). Bacha sirf 7 × 11. To 77 = 7 × 11 ✅, 11 se divisible.
- Yahi alternate difference hai: units digit “+” mein gaya, tens digit “−” mein. 7 − 7 = 0, to divisible.
- Hundreds: 100 = 99 + 1, aur 99 = 9 × 11. To 100 = 9 × 11 + 1: har hundred, 11 ke groups ke baad 1 zyada deta hai. Isliye hundreds digit “+” mein.
- Thousands: 1001 = 11 × 91 (check: 11 × 91 = 1001 ✅). 1000 = 1001 − 1: har thousand, 11 ke groups se 1 kam. Isliye thousands digit “−” mein.
- Pattern, right se: units +, tens −, hundreds +, thousands −, yaani + − + −. 11 ke groups to hamesha divisible hain; decision sirf in + aur − ke total se hota hai. Isiliye right se 1st, 3rd … digits ka sum aur 2nd, 4th … digits ka sum, aur dono ka difference dekhte hain.
11 ka diagram
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: 3716, 4 se divisible?
3716 ÷ 4? Last 2 digits: 16. 16 ÷ 4: 16 = 4 × 4 → ✅ divisible. Check: 3716 ÷ 4 = 929.
✔️ Normal method se check: 4 × 929 = 3716 ✅
Example 2: 53128, 8 se divisible?
53128 ÷ 8? Last 3 digits: 128. 128 ÷ 8: 128 = 8 × 16 → ✅ divisible. Check: 53128 ÷ 8 = 6641.
128 ko 3 baar half karo: 128 → 64 → 32 → 16 (har baar whole number aaya, 3 baar half ho gaya ✅).
✔️ Normal method se check: 8 × 6641 = 53128 ✅
Example 3: 7392, 11 se divisible?
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: 11 × 672 = 7392 ✅
Example 4: 2028 leap year hai? (Year ka number 4 se divisible?)
2028 ÷ 4? Last 2 digits: 28. 28 ÷ 4: 28 = 4 × 7 → ✅ divisible. Check: 2028 ÷ 4 = 507.
Haan, 2028 leap year hai: Dadi ka birthday!
✔️ Normal method se check: 4 × 507 = 2028 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
CharuMam ki smart trick: “Round part ko pehle hi pass kar do.” 100 mein 4 poora jaata hai, 1000 mein 8 poora jaata hai, isliye in round parts ko check karne ki zaroorat hi nahi; sirf bacha hua chhota part test karo. Aur 11 ke liye “ek kam, ek zyada” ki pair: 10 ek kam, 100 ek zyada. Isliye + aur − baari-baari.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| ÷ 4: remainder 0 hai ya nahi, ya remainder | ✅ Haan | Last 2 digits se. |
| ÷ 8 | ✅ Haan | Last 3 digits, 3 baar half. |
| ÷ 11 | ✅ Haan | Alternate sums ka difference. |
| Leap year check | ✅ Haan | Year ke last 2 digits (00 wale year mein ek extra rule hai, FAQ dekho). |
| ÷ 7, ÷ 13 | ❌ Nahi | Inke aasaan rules nahi; seedha divide karo. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: 4 ke liye sirf last digit
Sawaal: 26, 4 se divisible?
❌ Galat: “16 divisible hai aur iska bhi last digit 6, to haan.”
✅ Sahi: last 2 digits: 26 → 13. Sirf 1 half hua, 13 odd hai aur 2nd half nahi ho sakta → nahi. 26 = 4 × 6 + 2.
Mistake 2: 8 ke liye last 2 digits
Sawaal: 1124, 8 se divisible?
❌ Galat: “Last 2 digits 24 = 8 × 3, to haan.”
✅ Sahi: 8 ke liye last 3 digits: 124 → 62 → 31. Sirf 2 half hue, 31 odd hai aur 3rd half nahi ho sakta → nahi. 1124 = 8 × 140 + 4.
Mistake 3: 11 mein 22 ko “nahi” samjha
Sawaal: 918082, 11 se divisible?
❌ Galat: “Sums 3 aur 25, difference 22, 0 nahi to nahi.”
✅ Sahi: 22 bhi 11 ki table mein hai → divisible. 918082 ÷ 11 = 83462.
Mistake 4: 4 se divisible to 8 se bhi?
Sawaal: 12, 8 se divisible?
❌ Galat: “12, 4 se divisible hai, to 8 se bhi.”
✅ Sahi: 12 → 6 → 3 (odd, teesra half nahi) → nahi. 12 = 8 × 1 + 4. (Reverse sahi hai: 8 se divisible number 4 se bhi.)
🧠 Rule of thumb: 4: last 2 digits (2 baar half). 8: last 3 digits (3 baar half). 11: right se alternate digits ke sums ka difference 0, 11, 22 …
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal: “Haan/Nahi” aur kuch remainder wale. Target: 4 minute.
- 3716: 4 se divisible?
- 5318: 4 se divisible?
- 2028: 4 se divisible?
- 7100: 4 se divisible?
- 53128: 8 se divisible?
- 1124: 8 se divisible?
- 7000: 8 se divisible?
- 9136: 8 se divisible?
- 7392: 11 se divisible?
- 4825: 11 se divisible?
- 1342: 11 se divisible?
- 2332: 11 se divisible?
- 5318 ÷ 4 ka remainder = ?
- 9999 ÷ 4 ka remainder = ?
- 1234 ÷ 4 ka remainder = ?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- Haan (last 2 digits 16)
- Nahi (last 2 digits 18)
- Haan (last 2 digits 28)
- Haan (last 2 digits 00)
- Haan (last 3 digits 128)
- Nahi (last 3 digits 124)
- Haan (last 3 digits 000)
- Haan (last 3 digits 136)
- Haan (difference 11)
- Nahi (difference 7)
- Haan (difference 0)
- Haan (difference 0)
- 2
- 3
- 2
Kaise kiya (kuch): 7000: last 3 digits 000 → 8 se divisible. 9136: last 3 digits 136 → 68 → 34 → 17. Teeno half poore ho gaye; 17 odd hai par ab koi half baaki nahi → haan (9136 = 8 × 1142). 2332: (2 + 3) aur (3 + 2), difference 0 → haan. 9999 ÷ 4: 99 = 4 × 24 + 3 → remainder 3.
— 📌 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.
- 116 ÷ 4 ka remainder = ?
- 222 ÷ 4 ka remainder = ?
- 1000 ÷ 4 ka remainder = ?
- 3333 ÷ 4 ka remainder = ?
- 2024 ÷ 4 ka remainder = ?
- 2026 ÷ 4 ka remainder = ?
- 510 ÷ 4 ka remainder = ?
- 7777 ÷ 4 ka remainder = ?
- 4404 ÷ 4 ka remainder = ?
- 1918 ÷ 4 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
- 2
- 0
- 1
- 0
- 2
- 2
- 1
- 0
- 2
Dhyaan: sirf last 2 digits ÷ 4 ka remainder. 3333: 33 = 4 × 8 + 1 → 1. 2026: 26 = 4 × 6 + 2 → 2. 1918: 18 = 4 × 4 + 2 → 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. 3716, 4 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (c) Haan, last 2 digits 16 = 4 × 4
- (a) galat ❌: Digit sum 3 aur 9 ka test hai, 4 ka nahi.
- (b) galat ❌: Even hona sirf 2 ka test; 26 even hai par 4 se nahi.
- (c) sahi ✅: 16 → 8 → 4: do baar half poora → divisible (3716 ÷ 4 = 929).
- (d) galat ❌: 4 ke liye last 2 digits dekhte hain, sirf last digit nahi.
Q2. 8 se divisible hone ke liye kaunse digits dekhte hain?
Jawab aur reason dekho
Sahi jawab: (b) Last 3 digits
- (a) galat ❌: Digit sum 3 aur 9 ke liye.
- (b) sahi ✅: 1000 = 8 × 125, isliye thousands se left wala part hamesha divisible.
- (c) galat ❌: Last 2 digits 4 ke liye; 1124 mein 24 pass par 1124 fail.
- (d) galat ❌: Last digit 2, 5, 10 ke liye.
Q3. 7392, 11 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (a) Haan, difference 11 hai
- (a) sahi ✅: (2 + 3) = 5, (9 + 7) = 16, difference 11 → divisible (7392 ÷ 11 = 672).
- (b) galat ❌: 11 ke liye digit sum nahi, alternate sums ka difference.
- (c) galat ❌: Last digit 11 ke liye kuch nahi batata.
- (d) galat ❌: Even hona 2 ka test hai.
Q4. 1124, 8 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (d) Nahi, 124 → 62 → 31: 3rd half nahi ho sakta
- (a) galat ❌: Last 2 digits 8 ke liye kaafi nahi; last 3 chahiye.
- (b) galat ❌: 4 se divisible hona 8 ke liye kaafi nahi (12 jaisa).
- (c) galat ❌: 1124 even hai (last digit 4).
- (d) sahi ✅: Last 3 digits 124: teesra half poora nahi → remainder 4.
Q5. 5318 ÷ 4 ka remainder?
Jawab aur reason dekho
Sahi jawab: (c) 2
- (a) galat ❌: 3 kahin nahi aata; 18 ÷ 4 ka remainder 2.
- (b) galat ❌: 18 last 2 digits hain, remainder 4 se chhota hota hai.
- (c) sahi ✅: Last 2 digits 18 = 4 × 4 + 2 → remainder 2.
- (d) galat ❌: 0 tab jab last 2 digits 4 se divisible hon.
Q6. Kaunsa year leap year hoga (4 se divisible)?
Jawab aur reason dekho
Sahi jawab: (b) 2028
- (a) galat ❌: 18 = 4 × 4 + 2 → nahi.
- (b) sahi ✅: 28 = 4 × 7 → divisible.
- (c) galat ❌: 26 = 4 × 6 + 2 → nahi.
- (d) galat ❌: 30 = 4 × 7 + 2 → nahi.
Q7. 918082 ke alternate sums 3 aur 25 hain. Divisible by 11?
Jawab aur reason dekho
Sahi jawab: (a) Haan, difference 22
- (a) sahi ✅: 22 = 11 × 2, 11 ki table mein → divisible (918082 ÷ 11 = 83462).
- (b) galat ❌: 0 ke alawa 11, 22, 33 … bhi chalte hain.
- (c) galat ❌: Bada-chhota kuch nahi; difference dekhna hai.
- (d) galat ❌: Rule se pata chal jaata hai: difference 22.
Q8. 4 ka rule last 2 digits pe kyun chalta hai?
Jawab aur reason dekho
Sahi jawab: (d) 100 = 4 × 25, to hundreds hamesha divisible
- (a) galat ❌: Even hona reason nahi; 6 bhi even hai par uska aisa rule nahi.
- (b) galat ❌: 513 mein 4 nahi hai.
- (c) galat ❌: Yeh sach hai par rule ka reason nahi.
- (d) sahi ✅: Hundreds wala part hamesha 4 se divisible; sirf last 2 digits bachte hain.
Q9. Kaunsa number 11 se divisible hai?
Jawab aur reason dekho
Sahi jawab: (c) 2332
- (a) galat ❌: (3 + 3) aur (2 + 2): difference 2 → nahi.
- (b) galat ❌: (2 + 2) aur (3 + 3): difference 2 → nahi.
- (c) sahi ✅: (2 + 3) − (3 + 2) = 0 → divisible (2332 ÷ 11 = 212).
- (d) galat ❌: (3 + 3) aur (3 + 2): difference 1 → nahi.
Q10. Kaunsa statement sahi hai?
Jawab aur reason dekho
Sahi jawab: (b) 8 se divisible number 4 se bhi divisible
- (a) galat ❌: 16 aur 26 ka last digit same hai, par 26, 4 se divisible nahi.
- (b) sahi ✅: 8 = 4 × 2, to 8 ke groups, 4 ke groups mein bhi toot jaate hain.
- (c) galat ❌: 12, 4 se divisible hai par 8 se nahi.
- (d) galat ❌: 11 ke liye alternate digits ke sums ka difference dekhte hain.
— 📌 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. 73□6 ko 4 se divisible banana hai. □ kya-kya ho sakta hai?
Solution dekho
Last 2 digits □6. 4 ki table mein 6 pe khatam hone wale 2-digit numbers: 16, 36, 56, 76, 96. To □ = 1, 3, 5, 7, 9 (paanch options).
Check: 7316 ÷ 4 = 1829, 7396 ÷ 4 = 1849 ✅
Q2. Koi bhi 4-digit “mirror number” (jaise 2332, 4774, 5115) 11 se divisible hota hai. Kyun?
Solution dekho
Mirror number mein digits a, b, b, a. Right se 1st aur 3rd: a + b. 2nd aur 4th: b + a. Dono sums same → difference 0 → hamesha 11 se divisible. Jaise 4774 ÷ 11 = 434 ✅.
Q3. 237237 (koi 3-digit number do baar likha) 11 se divisible hai?
Solution dekho
1st, 3rd, 5th: 7 + 2 + 3 = 12. 2nd, 4th, 6th: 3 + 7 + 2 = 12. Difference 0 → haan. 237237 ÷ 11 = 21567 ✅. Aisa har “abcabc” number ke saath hota hai.
Q4. 1000 se chhota sabse bada number kaunsa hai jo 8 se divisible ho?
Solution dekho
1000 khud 8 se divisible (1000 = 8 × 125), to usse pichla 8 ki table wala: 1000 − 8 = 992. Test: 992 → 496 → 248 → 124, teen baar half poora ✅. 992 ÷ 8 = 124.
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | 4, 8, 11 ke rules, missing digit (73□6), leap year. |
| Maths Olympiad | Mirror numbers, “abcabc” numbers, sabse bada/chhota divisible number. |
| NTSE / MAT | Calendar (leap year) aur number series. |
| SSC / Bank (aage) | Number system ke sawaal: 4, 8, 11 se divisibility. |
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. 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: “4 ke liye 2, 8 ke liye 3, aur 11 ke liye baari-baari.” Har year ka number dekh kar socho leap year hai ya nahi. Agle post mein saare rules ka mix: tricky sawaal, missing digits aur 12, 15, 18 jaise combined rules! 🧮
— 📌 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)
Sirf last 3 digits: 128. 128 ko 3 baar half: 128 → 64 → 32 → 16, har baar whole number → 8 se divisible ✅. Baaki 53000 to hamesha 8 se divisible (1000 = 8 × 125).
| 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. Last 2 digits “04” ho to?
Matlab 4: 3704 → 4 ÷ 4 = 1, divisible. “00” matlab 0, woh bhi divisible.
2. 11 ke liye left se shuru karein to?
Difference ka jawab same aata hai (bas bada-chhota badal sakta hai). Habit ke liye right se shuru karo.
3. Year 1900 leap year tha?
Nahi! Jo year 00 pe khatam ho (1900, 2100), woh tabhi leap year jab 400 se divisible ho. 2000 tha (2000 = 400 × 5). 1900 = 400 × 4 + 300, yaani 300 bacha, isliye 1900 leap year nahi.
4. 8 ke liye 3 baar half long lagta hai?
Last 3 digits chhote hain; 3 baar half mann mein ho jaata hai. Ya 8 ki table yaad ho to seedha.
5. Agle post mein kya hai?
Post 17.4: tricky sawaal: saare rules ka mix, missing digit, 12, 15, 18 jaise combined rules aur dhokhe.
— 📌 charumam.com
🟢 Recap: Aaj Kya Seekha
- 4: last 2 digits 4 se divisible (100 = 4 × 25).
- 8: last 3 digits 8 se divisible (1000 = 8 × 125).
- Test: 4 ke liye 2 baar half, 8 ke liye 3 baar half. Half baaki ho aur odd aa jaaye → nahi (124 → 62 → 31). Saare half poore ho gaye → haan, chahe last number odd ho (136 → 68 → 34 → 17).
- 11: right se 1st, 3rd … ka sum aur 2nd, 4th … ka sum; difference 0, 11, 22 … → divisible.
- ÷ 4 aur ÷ 8 ka remainder bhi last digits se.
— charumam.com 🎯
👉 Aage Kya?
Ab Post 17.4: Divisibility ke tricky sawaal. Saare rules ek saath: 37□ ko 9 se divisible banao, 12 = 3 aur 4 dono, 15 = 3 aur 5 dono, aur dhokhe wale sawaal jaise “4 aur 2 se divisible, to 8 se bhi?”
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
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