Divisibility: 4, 8 Aur Digital Root | Class 8 Maths Book 1 Ch 5

📘 Class 8 · Ganita Prakash Book 1Ch 5 · Number PlayPost 6 / 7⏱ 25–30 min

Divisibility: 4, 8 Aur Digital Root 🌱

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:

100 ÷ 4 = ?
1000 ÷ 8 = ?
9 se remainder kaise nikaala tha?

Answers dekho

25 (poora bata)
125 (poora bata)
Digits ka jod

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 5 — Divisibility Rule: 9, 3 Aur 11 dekh lijiye.

— 📌 charumam.com

🎬 Kahani: Nani Ka Purana Hisaab-Check

Riya ne homework mein 47 × 38 = 1,796 likha. Nani ne bina calculator ek nazar dekha: “Galat hai, beta.” 👵

Riya: “Aapne kaise pakda?” Nani: “4 + 7 = 11 → 2. 3 + 8 = 11 → 2. 2 × 2 = 4. Tumhare jawab mein 1 + 7 + 9 + 6 = 23 → 5. 4 nahi, to galat!”

Yeh “nau ka niyam” hazaar saal purana hai. Aaj seekhenge yeh kyun chalta, aur kab dhokha deta. Saath mein 4, 8, 6, 24 ke rules ka “kyun” bhi. 🔍

— 📌 charumam.com

📖 Concept: Jo Hissa Pehle Se Bata Hai, Use Chhod Do

💡 Is post ki ek badi baat: Kisi divisor ke rule ka raaz: number ka jo hissa us divisor se pehle se bata hai, use chhod do; sirf bacha hua hissa dekho. 4 ke liye 100 bata hai, 8 ke liye 1000, 9 ke liye har place value ka “9 wala hissa”.

Step 1: 4 ka rule: aakhri do digits (kyun?)

7,316 = 73 × 100 + 16. Aur 100 = 4 × 25 → 73 × 100 pehle se 4 se bata.

Bacha 16 → 4 se bata → 7,316 bata ✅ Check: 4 × 1,829 = 7,316 ✅

WHY do digits? 10 to 4 se nahi bat-ta (2 bachta), par 100 bat-ta. Isliye tens + units dono dekhne padte.

Step 2: 8 ka rule: aakhri teen digits

1000 = 8 × 125. To 7,316 = 7 × 1000 + 316 → sirf 316 dekho: 316 ÷ 8 = 39.5 → nahi.

Pattern: 2 → 1 digit (10 bata), 4 → 2 digit (100), 8 → 3 digit (1000), 16 → 4 digit (10000 = 16 × 625)!

Step 3: 6 ka rule: 2 aur 3 dono

6 = 2 × 3, aur 2, 3 ka koi common factor nahi. To dono se bate → LCM 6 se bate (Post 3 — Hamesha, Kabhi-Kabhi Ya Kabhi Nahi?).

186: even ✅, 1 + 8 + 6 = 15 → 3 se ✅ → 6 se ✅ Check: 6 × 31 = 186 ✅

Step 4: 24 ka rule: 4 aur 6 nahi, 3 aur 8!

12 → 4 se bhi, 6 se bhi… par 24 se nahi! Kyun? 4 = 2 × 2, 6 = 2 × 3: common 2. LCM sirf 12.

24 = 2 × 2 × 2 × 3 = 8 × 3 (koi common factor nahi). 5,736: 736 ÷ 8 = 92 ✅, 5 + 7 + 3 + 6 = 21 ✅ → 24 se ✅ Check: 24 × 239 = 5,736 ✅

Kab kaunsa? Bade divisor ko co-prime tukdon mein todo (6 = 2 × 3, 24 = 8 × 3, 36 = 4 × 9, 44 = 4 × 11). Common factor wale tukde (4 × 6) dhokha dete.

Step 5: Digital root aur uska raaz

Digits baar-baar jodo jab tak ek digit na bache: 58,274 → 26 → 8.

Digital root = 9 se remainder (aur 9 ke multiples ka root 9). Kyun? Post 5 — Divisibility Rule: 9, 3 Aur 11 ka tark: har jod pe 9 ka remainder same rehta.

Lagaataar numbers ke roots: 1, 2, 3 … 9, 1, 2 … ghoomte rehte, jaise 9 wali ghadi.

Step 6: Guna check (casting out nines)

47 × 38: roots 2 aur 2 → 2 × 2 = 4. Sahi jawab 1,786: 1 + 7 + 8 + 6 = 22 → 4 ✅. Riya ka 1,796 → 5 ❌.

WHY kaam karta? Post 4 — Remainder Ka Khel: guna ka remainder = remainders ka guna. Root = 9 ka remainder.

Dhyaan: 1,768 (digits ulat-palat) ka root bhi 4! Check galti pakad sakta, par har galti nahi.

🪜 Atak Gaye? Aise Socho

2/5/10: aakhri digit. 4: aakhri do. 8: aakhri teen.
3/9: digit jod. 11: units se + − + −.
Bada divisor: co-prime tukdon mein todo.
Digital root = 9 ka remainder (9 = 0 jaisa).
Guna check: roots ka guna ka root = jawab ka root?

— 📌 charumam.com

🔐 Deep Concept: “Pehle Se Bata Hissa” Ek Hi Idea Hai Har Rule Mein

Saare rules ek line mein:

Divisor	Pehle se bata hissa	Kya dekho
2, 5, 10	10, 100, 1000…	Aakhri digit
4, 25	100, 1000…	Aakhri 2 digit
8, 125	1000…	Aakhri 3 digit
9, 3	9, 99, 999 wale hisse	Digit jod
11	11, 99, 1001 wale hisse	+ − + − jod

Ek baar yeh dekh liya to 25 ka rule bhi aap khud bana sakte: 100 = 25 × 4, to aakhri do digits 00, 25, 50, 75! 🎯

Aryabhata II (lagbhag 950 CE) ne Mahāsiddhānta mein digital root se hisaab check karne ka zikr kiya. Hazaar saal pehle ka “error check”!

Unifying idea: poore chapter mein ek hi soch: bade number ko “bata hua hissa + bacha hissa” mein todo, sirf bache se faisla karo.

sirf charumam.com readers ke liye 🔐

✏️ Solved Examples

Example 1 (Easy): 3,47,528 kya 8 se bata?

Aakhri teen: 528 ÷ 8 = 66 → haan.

Check: 8 × 43,441 = 3,47,528 ✅

📌 charumam.com

Example 2 (Medium): 9,276 kya 36 se bata?

36 = 4 × 9 (co-prime). 76 ÷ 4 = 19 ✅; 9 + 2 + 7 + 6 = 24 ❌ (9 se nahi) → nahi.

Check: 36 × 257 = 9,252, 24 bache ✅

📌 charumam.com

Example 3 (NCERT style): 600 se 700 ke beech digital root 5 wale numbers?

600 ka root 6 → 608 ka root 5; phir har 9 aage: 608, 617, 626, …, 698 (11 numbers).

Check: 6 + 9 + 8 = 23 → 5 ✅, (698 − 608)/9 + 1 = 11 ✅

📌 charumam.com

Nani ka check: 2 × 2 = 4, par 1,796 ka root 5 → galti pakdi. Riya ne dobara kiya: 1,786. “Nani, aap to hazaar saal purana computer ho!” Nani: “Aryabhata wala!” 😄

— 📌 charumam.com

🏠 Divisibility Aur Digital Root Roz Ki Zindagi Mein

📅 Leap year: saal ke aakhri do digits 4 se bate? (2028 → 28 ✅). 4 wala rule!
🧾 Hisaab ki kitaab: munim ji digital root se lambe jod check karte.
📦 Carton: 24 bottle ka carton: total bottles 3 aur 8 dono se bate?
🎰 Lucky number: log janamtithi ka digital root nikaalte; maths mein yeh bas 9 ka remainder hai!
💻 Data check: computer “checksum” se file ki galti pakadte: wahi idea.

Leap year ka asli niyam thoda aur hai: 100 se bate saal (1900) leap nahi, par 400 se bate (2000) leap. Divisibility hi divisibility!

— 📌 charumam.com

🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
4 / 8	✅ Aakhri 2 / 3 digit	100 / 1000 bate
6	✅ 2 aur 3	Co-prime
24	✅ 3 aur 8	4 aur 6 ❌
Digital root	✅ 9 ka remainder	Lagaataar ghoomta
Guna check	✅ Roots ka guna	Har galti nahi pakadta

— 📌 charumam.com

❌→✅ Worked Mistake

Sawaal: 36 kya 24 se bata? (“4 aur 6 dono se bate to 24 se.”)

❌ Galat: 36 ÷ 4 = 9 ✅, 36 ÷ 6 = 6 ✅ → “24 se bata.” Par 36 ÷ 24 = 1.5!

✅ Sahi: 3 aur 8 dekho: 36 ÷ 8 = 4.5 ❌ → nahi. Check: 24 × 1 = 24, 24 × 2 = 48: 36 beech mein ✅

Doosri common galti: 4 ke liye sirf aakhri digit dekhna: 14 even hai par 4 se nahi.

🧠 Rule of thumb: Bade divisor ko aise tukdon mein todo jinka koi common factor na ho.

sirf charumam.com pe milenge!

📝 10 MCQs

Q1. 5,132 kya 4 se?

(a) Haan

(b) Nahi

(c) Sirf 2 se

(d) Pata nahi

Solution dekho

Sahi jawab: (a)

(a) sahi: 32 ÷ 4 = 8, to bata hai ✅
(b) galat: Aakhri do digit 32, 4 se bate.
(c) galat: 4 se bhi bata, sirf 2 nahi.
(d) galat: Aakhri do digit se pakka pata.

Q2. 2,516 kya 8 se?

(a) Haan

(b) Nahi

(c) Sirf 4 se bhi nahi

(d) Hamesha

Solution dekho

Sahi jawab: (b)

(a) galat: 516 ÷ 8 = 64.5, poora nahi.
(b) sahi: 516 ÷ 8 poora nahi → nahi ✅
(c) galat: 16 ÷ 4 = 4, to 4 se bata hai.
(d) galat: Hamesha jaisa kuch nahi yahan.

Q3. 24 ke liye kaunsa jodi sahi test?

(a) 4 aur 6

(b) 3 aur 8

(c) 2 aur 12

(d) 4 aur 3

Solution dekho

Sahi jawab: (b)

(a) galat: 12 dono se bata, 24 se nahi.
(b) sahi: 3 × 8, common factor nahi ✅
(c) galat: 2 aur 12 ka common 2 hai.
(d) galat: 4 × 3 = 12 hi hai, 24 nahi.

Q4. 7,295 ka digital root?

(a) 5

(b) 14

(c) 23

(d) 4

Solution dekho

Sahi jawab: (a)

(a) sahi: 7 + 2 + 9 + 5 = 23 → 5 ✅
(b) galat: 14 kahaan se aaya, dobara jodo.
(c) galat: 23 ko phir jodo: 2 + 3 = 5.
(d) galat: 4 ka hisaab galat hai.

Q5. 9 ke multiple ka digital root?

(a) 0

(b) 9

(c) 1

(d) 3

Solution dekho

Sahi jawab: (b)

(a) galat: Digital root 0 nahi aata (positive).
(b) sahi: Hamesha 9 hi aata hai ✅
(c) galat: 1 to 10, 19 jaison ka.
(d) galat: 3 kuch multiples of 3 ka.

Q6. Digital root 7 wale number + 10 ka root?

(a) 8

(b) 17

(c) 7

(d) 1

Solution dekho

Sahi jawab: (a)

(a) sahi: 10 ka root 1 → 7 + 1 = 8 ✅
(b) galat: 17 ko phir jodo: 1 + 7 = 8.
(c) galat: 10 jodne se root badalta hai.
(d) galat: 1 to sirf 10 ka root hai.

Q7. 6 ka test kaunsa?

(a) 2 aur 3 dono

(b) Sirf 3

(c) Digit jod 6

(d) Aakhri digit 6

Solution dekho

Sahi jawab: (a)

(a) sahi: 2 aur 3 co-prime, LCM 6 ✅
(b) galat: 15 ko 3 se, par 6 se nahi.
(c) galat: 15 ka jod 6, par 6 se nahi bata.
(d) galat: 16 ka aakhri 6, par 6 se nahi.

Q8. Guna check kaunsi galti nahi pakadta?

(a) Digits ki adla-badli

(b) Ek digit galat

(c) +1 ki galti

(d) Kuch bhi galat

Solution dekho

Sahi jawab: (a)

(a) sahi: 1786 ↔ 1768: root same ✅
(b) galat: Ek digit badla to root badalta.
(c) galat: +1 se root ek badalta hai.
(d) galat: Bahut si galtiyaan pakad leta hai.

Q9. 100 ÷ 4 pe kya, aur isliye 4 ka rule?

(a) 25, aakhri 2 digit

(b) 25, aakhri digit

(c) 20, aakhri 2

(d) Bachta 0, digit jod

Solution dekho

Sahi jawab: (a)

(a) sahi: 100 bata → 2 digit kaafi ✅
(b) galat: 10 4 se nahi bat-ta, 1 kaafi nahi.
(c) galat: 100 ÷ 4 = 25, 20 nahi.
(d) galat: Digit jod 9 ke liye hai.

Q10. 36 ka sahi test?

(a) 4 aur 9

(b) 6 aur 6

(c) 3 aur 12

(d) 2 aur 18

Solution dekho

Sahi jawab: (a)

(a) sahi: Co-prime hain, guna 36 ✅
(b) galat: 6 aur 6 ek hi baat hai.
(c) galat: 3 aur 12 ka common 3 hai.
(d) galat: 2 aur 18 ka common 2 hai.

— 📌 charumam.com

❓ FAQs (Bachchon Ke Asli Sawaal)

1. 16 ka rule?

10000 = 16 × 625, to aakhri 4 digits dekho.

2. 25 ka rule?

100 = 25 × 4 → aakhri do digits 00, 25, 50, 75.

3. Digital root aur parity ka rishta?

Seedha nahi: 11 (root 2) odd, 2 even. Root sirf 9 ka remainder batata, 2 ka nahi.

4. Kya digital root se 3 ka pata?

Haan: root 3, 6, 9 → 3 se bata. Kyunki root = 9 ka remainder.

5. Jod check bhi ho sakta?

Haan: roots ka jod ka root = jawab ka root. Ghataav mein bhi.

6. Har number ke liye rule hai?

100 tak har number ke liye hai, par kuch lambe. Wahi “bata hissa” soch.

— 📌 charumam.com

🏋️ Practice Zone (12 Sawaal)

Q1. Easy 4,628 kya 4 se?

Solution dekho

28 → haan. Check: 4 × 1,157 ✅

Q2. Easy 9,124 kya 8 se?

Solution dekho

124 ÷ 8 = 15.5 → nahi. Check: 8 × 1,140 = 9,120, 4 bache ✅

Q3. Easy 534 kya 6 se?

Solution dekho

Even ✅, 12 ✅ → haan. Check: 6 × 89 ✅

Q4. Easy 86,402 ka digital root?

Solution dekho

20 → 2. Check: 86,402 = 9 × 9,600 + 2 ✅

Q5. Medium 3,312 kya 24 se?

Solution dekho

312 ÷ 8 = 39 ✅, 9 ✅ → haan. Check: 24 × 138 ✅

Q6. Medium Guna 56 × 73 = 4,098 sahi? Root check?

Solution dekho

Roots 2 × 1 = 2; 4,098 → 21 → 3 ❌ → galat (sahi 4,088 → 2). Check ✅

Q7. Medium 7a32 8 se bate. a?

Solution dekho

a32 ÷ 8: 032 ✅, 232 ✅, 432 ✅… 32 + 100a, 100a ke liye a even → 0, 2, 4, 6, 8. Check: 7,432 = 8 × 929 ✅

Q8. Medium 12 lagaataar numbers (50–61) ke digital roots?

Solution dekho

5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7. Check: ghadi ki tarah ghoomte ✅

Q9. NCERT style 9a + 36b + 22 ka digital root?

Solution dekho

9 ke multiple chhodo → 22 → 4. Check: a = b = 1 → 67 → 13 → 4 ✅

Q10. NCERT style 6-digit 52a10b 44 se bate (a, b digits)?

Solution dekho

4: 0b → b = 0, 4, 8. 11: b − 0 + 1 − a + 2 − 5 = b − a − 2 = 0 ya ±11 → (a, b) = (2, 4), (6, 8), (9, 0). Check: 522104 = 44 × 11866 ✅, 529100 = 44 × 12025 ✅

Q11. NCERT style 45,000 aur 45,100 ke beech 36 ke multiples?

Solution dekho

45,000 = 36 × 1250 → 45,036, 45,072 → 2. Check ✅

Q12. NCERT style 8 ka multiple, 3 ka multiple nahi: aisa 24 se bata?

Solution dekho

Kabhi nahi: 24 = 8 × 3, 3 zaroori. Check ✅

Answer Key — charumam.com exclusive

🧠 HOTS (Sochne Wale Sawaal)

Q1. Kya digital root se 11 ki galti pakad sakte?

Solution dekho

Nahi seedha: root 9 ka remainder hai. 11 ke liye + − wala jod. Check ✅

Q2. 11 baar 11 jodte jao (5, 16, 27…). Digital roots?

Solution dekho

Har baar 2 badhta: 5, 7, 9, 2, 4… sab 9 values ghoom ke. Check: 16 → 7 ✅

Q3. 8-digit number ka root 5. 10 zyada ka root?

Solution dekho

5 + 1 = 6. Check: 10 ka root 1 ✅

Q4. 3 lagaataar numbers: pehla 2 ka, doosra 3 ka, teesra 4 ka multiple. Sabse chhote?

Solution dekho

2, 3, 4; phir har 12 pe: 14, 15, 16. Check: 14 ÷ 2, 15 ÷ 3, 16 ÷ 4 ✅

sirf charumam.com pe milenge!

🏆 Olympiad Corner

Q1. 2025 ka digital root?

Solution dekho

9 (45² = 2025!).

Q2. Sabse chhota 5-digit 24 ka multiple?

Solution dekho

10,008.

Q3. Kisi perfect square ka digital root kabhi 2 ho sakta?

Solution dekho

Nahi: squares ke roots 1, 4, 7, 9.

Q4. 9876a 8 se bate. a?

Solution dekho

76a ÷ 8 → a = 0 ya 8.

sirf charumam.com readers ke liye 🔐

💗 CharuMam Motivation

Nani ka hazaar saal purana trick aaj aapka hai, aur uske “kyun” ke saath! Har rule ke peeche ek hi soch: bata hissa chhodo, bacha dekho. 🌟

— 📌 charumam.com

🟢 Master Recap

💡 Chaabi: Kisi divisor ke rule ka raaz: number ka jo hissa us divisor se pehle se bata hai, use chhod do; sirf bacha hua hissa dekho. 4 ke liye 100 bata hai, 8 ke liye 1000, 9 ke liye har place value ka “9 wala hissa”.
4: aakhri 2 digit (100 bata); 8: aakhri 3 (1000 bata).
6 = 2 × 3; 24 = 3 × 8; 36 = 4 × 9.
Common factor wale tukde dhokha dete (4 × 6).
Digital root = 9 ka remainder.
Guna check: roots ka guna.
Har galti nahi pakadta (adla-badli).

— charumam.com 🎯

👉 Aage Kya?

Ab in sab tricks ko paheliyon mein lagao: letters mein chhupe digits! Agla post: Post 7 — Cryptarithm: Digits In Disguise.

← Pichla: Post 5 — Divisibility Rule: 9, 3 Aur 11 📚 Chapter 5 ke saare posts Agla: Post 7 — Cryptarithm: Digits In Disguise →

Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Nani Ka Purana Hisaab-Check

📖 Concept: Jo Hissa Pehle Se Bata Hai, Use Chhod Do

Step 1: 4 ka rule: aakhri do digits (kyun?)

Step 2: 8 ka rule: aakhri teen digits

Step 3: 6 ka rule: 2 aur 3 dono

Step 4: 24 ka rule: 4 aur 6 nahi, 3 aur 8!

Step 5: Digital root aur uska raaz

Step 6: Guna check (casting out nines)

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: “Pehle Se Bata Hissa” Ek Hi Idea Hai Har Rule Mein

✏️ Solved Examples

🏠 Divisibility Aur Digital Root Roz Ki Zindagi 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?

You may also like