Divisibility Tricks Ka Raaz | Class 8 Maths Book 2 Ch 6

📘 Class 8 · Ganita Prakash Book 2Ch 6 · Algebra PlayPost 6 / 7⏱ 25–30 min

Divisibility Tricks Ka Raaz 🔍

IS POST MEIN

Pehle Yeh Aata Hai?Kahani Concept Deep Concept Examples Real Life Galti 10 MCQs FAQs Practice HOTS Olympiad Recap

✅ Pehle Yeh Aata Hai?

Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:

74 − 47 = ?
27 ÷ 9 = ?
Do-digit “ab” ka value?

Answers dekho

27
3
10a + b

Teeno sahi? Badhiya! Place value wala algebra: Post 5.

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🎬 Kahani: Ulta Number, Seedha Jawab

Rohan ne kaha: “Koi bhi do-digit number lo (dono digits alag). Use ulta karo. Bade mein se chhota ghatao. Ab 9 se bhaag do.”

74 → 47 → 27 → 27 ÷ 9 = 3. 91 → 19 → 72 → 8. 52 → 25 → 27 → 3. Har baar poora bat gaya, sheshfal 0! 🔍

Aur bhaagfal kya batata hai? Chalo algebra ki torch jalaate hain.

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📖 Concept: Place Value Ka Algebra

Step 1: Ulta karke ghatao → 9 se divide

Number = 10a + b, ulta = 10b + a. Farak = (10a + b) − (10b + a) = 9a − 9b = 9(a − b). Hamesha 9 ka multiple! Bhaagfal = digits ka farak (74: 7 − 4 = 3 ✅).

Step 2: Ulta karke jodo → 11 se divide

(10a + b) + (10b + a) = 11a + 11b = 11(a + b). 47 + 74 = 121 = 11 × 11 ✅ (4 + 7 = 11).

Step 3: Teen-digit ghumaao (cyclic)

abc + bca + cab = 111a + 111b + 111c = 111(a + b + c). 111 = 3 × 37, to jod hamesha 37 aur 3 se divide! 123 + 231 + 312 = 666 = 111 × 6 ✅

Step 4: abcabc ka jaadu

Teen-digit number do baar likho: 123123. Ise 7, phir 11, phir 13 se bhaag do: wapas 123! Kyun? abcabc = abc × 1000 + abc = abc × 1001, aur 1001 = 7 × 11 × 13. 🎩

Step 5: Jab a < b?

Phir ulta number bada; bade se chhota ghatao: 9(b − a). Niyam wahi.

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🔐 Deep Concept: Divisibility Ke Purane Niyam Kyun Chalte Hain?

9 ka niyam (digits ka jod 9 se bate) bhi place value se: 10 = 9 + 1, 100 = 99 + 1. To 100a + 10b + c = 99a + 9b + (a + b + c). Pehla hissa 9 se bat-ta hai, to poora tabhi jab a + b + c bate!

11 ka niyam (alternate digits ka farak) bhi aise hi: 10 = 11 − 1. Algebra har purane “rule” ka kyun bata deta hai. 🔐

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✏️ Solved Examples

Example 1 (Easy): 83 ulta-ghatao, 9 se bhaag. Bhaagfal?

83 − 38 = 45, ÷ 9 = 5 = 8 − 3.

WHY sahi hai? 9(a − b) ✅

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Example 2 (Medium): 56 + 65 kis se divide aur bhaagfal?

121 = 11 × 11 (5 + 6).

WHY sahi hai? 11(a + b) ✅

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Example 3 (NCERT style): abc = 245: abc + bca + cab, aur 37 se bhaag.

245 + 452 + 524 = 1221 = 111 × 11 → 1221 ÷ 37 = 33.

WHY sahi hai? 3 × 11 = 33 ✅

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Rohan ka mantra: “10a + b likho, raaz khud bolega!” 🎉

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🏠 Real Life Mein

🔢 Ginti check karna (9 se kaatna, purane hisaab mein).
🔐 ISBN, barcode mein check digits isi tarah ke niyam.
🎩 Party trick (abcabc).
🧮 Mental maths tez.

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Ulta − seedha	✅ 9(a − b)	9 ka multiple
Ulta + seedha	✅ 11(a + b)	11 ka multiple
abc + bca + cab	✅ 111(a+b+c)	3 aur 37
abcabc	✅ abc × 1001	7, 11, 13

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❌→✅ Worked Mistake

Sawaal: 62 + 26 = 88. “Yeh 9 se bat-ta hai kyunki ulta kiya.”

❌ Galat: Jod ko 9 wala niyam lagaaya.

Galti kya hai? 9 wala niyam ghatane pe hai ❌

✅ Sahi: Jod = 11(a + b): 88 = 11 × 8 ✅

Yaad rakho: Ghatao → 9, jodo → 11.

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📝 10 MCQs

Q1. (10a + b) − (10b + a)?

(a) 9(a − b)

(b) 11(a − b)

(c) 9(a + b)

(d) a − b

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Nahi.
(c) galat: Nahi.
(d) galat: Nahi.

Q2. 74 − 47 ÷ 9?

(a) 3

(b) 27

(c) 7

(d) 4

Solution dekho

Sahi jawab: (a)

(a) sahi: 7 − 4 ✅
(b) galat: Farak.
(c) galat: Nahi.
(d) galat: Nahi.

Q3. Jod (10a+b)+(10b+a)?

(a) 11(a + b)

(b) 9(a + b)

(c) 10(a + b)

(d) a + b

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Nahi.
(c) galat: Nahi.
(d) galat: Nahi.

Q4. abc + bca + cab?

(a) 111(a+b+c)

(b) 100(a+b+c)

(c) 3(a+b+c)

(d) abc

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Nahi.
(c) galat: Nahi.
(d) galat: Nahi.

Q5. 111 = ?

(a) 3 × 37

(b) 11 × 11

(c) 7 × 13

(d) 3 × 33

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: 121.
(c) galat: 91.
(d) galat: 99.

Q6. 1001 = ?

(a) 7 × 11 × 13

(b) 9 × 111

(c) 10 × 100

(d) 7 × 7 × 13

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: 999.
(c) galat: 1000.
(d) galat: 637.

Q7. 123123 ÷ 7 ÷ 11 ÷ 13?

(a) 123

(b) 1001

(c) 231

(d) 0

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Nahi.
(c) galat: Nahi.
(d) galat: Nahi.

Q8. 47 + 74 = 121 = 11 × ?

(a) 11

(b) 4

(c) 7

(d) 121

Solution dekho

Sahi jawab: (a)

(a) sahi: 4 + 7 ✅
(b) galat: Nahi.
(c) galat: Nahi.
(d) galat: Nahi.

Q9. 9 ka divisibility niyam kyun?

(a) 10 = 9 + 1

(b) Jaadu

(c) Sanyog

(d) Rule

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Nahi.
(c) galat: Nahi.
(d) galat: Wajah hai.

Q10. 245 + 452 + 524 ÷ 37?

(a) 33

(b) 11

(c) 37

(d) 1221

Solution dekho

Sahi jawab: (a)

(a) sahi: Sahi ✅
(b) galat: Nahi.
(c) galat: Nahi.
(d) galat: Jod.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Digits same hon (jaise 55)?

Farak 0, jo 9 se bat-ta (0 ÷ 9 = 0).

2. Teen-digit ulta-ghatao?

abc − cba = 99(a − c): 99 se bat-ta!

3. abcabc mein abc 3-digit hi?

Haan, warna 1001 wala niyam alag.

4. 37 kyun khaas?

111 ka factor; isliye 111, 222, … 999 sab 37 se bat-te.

5. Kya yeh har number pe?

Haan, algebra ne har case saabit kiya.

6. Purane niyam yaad nahi?

Algebra se khud bana lo!

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy 62 − 26 ÷ 9?

Solution dekho

Q2. Easy 38 + 83 ÷ 11?

Solution dekho

11.

Q3. Easy 95 − 59?

Solution dekho

36 = 9 × 4.

Q4. Easy 456456 ÷ 1001?

Solution dekho

456.

Q5. Medium abc − cba ka formula?

Solution dekho

99(a − c).

Q6. Medium 731 − 137?

Solution dekho

594 = 99 × 6.

Q7. Medium 135 + 351 + 513?

Solution dekho

999 = 111 × 9.

Q8. NCERT style Bhaagfal ka rishta digits se (ulta-ghatao)?

Solution dekho

Digits ka farak.

Q9. NCERT style Jod hamesha 11 se kyun?

Solution dekho

11(a + b).

Q10. Medium abc + bca + cab 3 se hamesha?

Solution dekho

Haan, 111 = 3 × 37.

Q11. Medium ab + ba = 154. a + b?

Solution dekho

154 ÷ 11 = 14.

Q12. NCERT style abcabc ÷ 7 ÷ 11 ÷ 13 kyun abc?

Solution dekho

abcabc = abc × 1001.

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🧠 HOTS (Sochne Wale Sawaal)

Q1. ab − ba = 54. Kaun se numbers?

Solution dekho

9(a − b) = 54 → a − b = 6: 71, 82, 93 (aur 60, jiska ulta 06 = 6).

Q2. Kya ab + ba kabhi 9 se bhi bat sakta?

Solution dekho

Haan, jab a + b = 9 (jaise 18 + 81 = 99).

Q3. abcd − dcba ka factor?

Solution dekho

999(a − d) + 90(b − c) = 9[111(a−d) + 10(b−c)] → 9 se.

Q4. Ulta-ghatao 2 baar karo (27 → 72 − 27 = 45 → 54 − 45 = 9)?

Solution dekho

Aakhir 9 ya 0 ki taraf; mazedaar khoj karo!

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🏆 Olympiad Corner

Q1. Ek do-digit number ka ulta usse 36 zyada, digits ka jod 10. Number?

Solution dekho

b − a = 4, a + b = 10 → 37.

Q2. abcabc 77 se bat-ta?

Solution dekho

Haan, 1001 = 7 × 11 × 13, to 77 bhi.

Q3. Kitne do-digit numbers jinka ulta-jod 121?

Solution dekho

a + b = 11: 29, 38, 47, 56, 65, 74, 83, 92 → 8.

Q4. 111…1 (6 ones) 7 se?

Solution dekho

111111 = 111 × 1001 → haan.

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💗 CharuMam Motivation

Rohan ki trick jaadu lagti thi, par ek line ke algebra ne sach dikha diya. 🔍

Jab bhi koi “niyam” rato, ek baar “kyun?” zaroor poochho. Samjha hua niyam kabhi nahi bhoolta! 🧠

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🟢 Master Recap

Ulta-ghatao: 9(a − b); bhaagfal = digits ka farak.
Ulta-jodo: 11(a + b).
abc + bca + cab = 111(a + b + c) → 3 aur 37.
abcabc = abc × 1001 = abc × 7 × 11 × 13.
9 aur 11 ke niyam bhi place value se.

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👉 Aage Kya?

Chapter ka aakhri post: algebra se paheliyaan: phool aur mandir, sir aur pair, Karim aur jinn! 🧞

← Pichla: Post 5 — Sabse Bada Guna 📚 Chapter 6 ke saare posts Agla: Post 7 — Algebra Se Paheliyaan Suljhao →

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✅ Pehle Yeh Aata Hai?

🎬 Kahani: Ulta Number, Seedha Jawab

📖 Concept: Place Value Ka Algebra

Step 1: Ulta karke ghatao → 9 se divide

Step 2: Ulta karke jodo → 11 se divide

Step 3: Teen-digit ghumaao (cyclic)

Step 4: abcabc ka jaadu

Step 5: Jab a < b?

🔐 Deep Concept: Divisibility Ke Purane Niyam Kyun Chalte Hain?

✏️ Solved Examples

🏠 Real Life Mein

🧭 Kab Use Karein, Kab Nahi

❌→✅ Worked Mistake

📝 10 MCQs

❓ FAQs (Bachchon Ke Asli Sawaal)

🏋️ Practice Zone (12 Sawaal)

🧠 HOTS (Sochne Wale Sawaal)

🏆 Olympiad Corner

💗 CharuMam Motivation

🟢 Master Recap

👉 Aage Kya?

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