Hamesha, Kabhi-Kabhi Ya Kabhi Nahi? | Class 8 Maths Book 1 Ch 5

📘 Class 8 · Ganita Prakash Book 1Ch 5 · Number PlayPost 3 / 7⏱ 25–30 min

Hamesha, Kabhi-Kabhi Ya Kabhi Nahi? 🧐

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:

Multiple ka matlab?
12 aur 18 ka LCM?
8a + 8b = ?

Answers dekho

Kisi number ka guna (8, 16, 24… 8 ke)
36
8(a + b)

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 2 — Even-Odd Ka Khel: Parity dekh lijiye.

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🎬 Kahani: Class Ka Debate: “Yeh Hamesha Sach Hai!”

Ritu ne board pe likha: “Jo number 4 se bhi bate aur 6 se bhi, woh 24 se bhi batega.” Usne 24, 48, 72 dikhaaye. “Dekha? Hamesha!” 📢

Sameer khada hua: “12?” 12 ÷ 4 = 3, 12 ÷ 6 = 2… par 12 ÷ 24? Nahi! Class mein sannata.

Ritu ke teen udaharan ek pal mein gir gaye, sirf ek counter-example se. Aaj seekhenge daavon ko kaise parkhte hain. ⚖️

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📖 Concept: Teen Dibbe, Teen Tarah Ke Saboot

💡 Is post ki ek badi baat: Hamesha saabit karne ke liye algebra (ya chitra) chahiye, udaharan kaafi nahi. Kabhi nahi ke liye bhi tark chahiye. Kabhi-kabhi ke liye bas ek “haan” aur ek “na” udaharan. Aur “hamesha” todne ke liye ek counter-example kaafi.

Step 1: Daava 1: 6 dono ko baante to jod ko bhi

6a + 6b = 6(a + b) → hamesha. Farak bhi: 6a − 6b = 6(a − b).

Chitra: 6-6 ki pankti mein a pankti + b pankti = (a + b) poori panktiyaan. Koi adhoori pankti nahi!

Niyam: a, M aur N dono ko baante → M + N aur M − N ko bhi.

Step 2: Daava 2: jod 10 ka multiple → dono number bhi?

30 = 20 + 10 ✅ par 30 = 17 + 13 ❌ → kabhi-kabhi.

WHY ulta kaam nahi karta? Do adhoori panktiyaan (7 aur 3 bache) milke poori ban sakti hain. Niyam ek taraf chalta, ulta nahi.

Step 3: Daava 3 aur 4: multiples aur factors

5 se bate number ke saare multiples 5 se batenge: 5k × m = 5(km) → hamesha.

18 se bate number 1, 2, 3, 6, 9, 18 sab se batenge: 18m = 2 × 9 × m = 3 × 6 × m → hamesha.

Par 5 se bate → 15 se bhi? 30 ✅, 20 ❌ → kabhi-kabhi. (Factor neeche jaata hai, multiple upar nahi.)

Step 4: Daava 5: do numbers se bate → kis se pakka?

4 aur 10 dono se bate → 40 se? 20 ❌ → kabhi-kabhi. Par LCM 20 se hamesha ✅

5 aur 8 dono se → 40 se hamesha, kyunki 5 aur 8 ka koi common factor nahi (LCM = 40).

WHY LCM? Number ke prime factors mein 4 = 2 × 2 bhi chahiye aur 10 = 2 × 5 bhi. Common 2 ek hi baar gina jaata: 2 × 2 × 5 = 20.

Step 5: Daava 6: kabhi nahi ka saboot

“Teen lagaataar odd numbers ka jod 6 ka multiple hai.”

(2n − 1) + (2n + 1) + (2n + 3) = 6n + 3 → hamesha odd → 6 ka multiple (even) kabhi nahi. Par 3 ka multiple hamesha: 3(2n + 1)!

Kab kaunsa saboot? Hamesha/kabhi nahi: algebra ya parity. Kabhi-kabhi: do udaharan. Chitra: jab “panktiyon” mein sochna aasaan ho.

🪜 Atak Gaye? Aise Socho

Pehle 3-4 numbers try karo: koi “na” mila?
Mila → hamesha nahi. Ek “haan” bhi mila → kabhi-kabhi.
Sab “haan” → algebra se likho (6a, 6b…), saboot dhundho.
Sab “na” → parity ya tark se dikhao kyun kabhi nahi.
Do numbers se bhaag ho to LCM yaad karo.

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🔐 Deep Concept: Udaharan Kyun Kaafi Nahi? (Ek Kaala Hans Ki Kahani)

Europe ke log sadiyon tak maante rahe: “Har hans safed hai.” Laakhon safed hans dekhe the! Phir Australia mein ek kaala hans mila, aur daava gir gaya. 🦢

Maths mein bhi: 1000 udaharan “hamesha” saabit nahi karte, par 1 counter-example “hamesha” tod deta. Isliye algebra: 6a + 6b = 6(a + b) har a, b ke liye ek saath bolta, anginat udaharan ek line mein.

Ek-taraf niyam: “a M aur N ko baante → a M + N ko baante” sach, par ulta nahi. Ch 4 ke ulte (converse) jaisa: niyam ki disha dekho.

Unifying idea: daava = disha + shart. Har baar poochho: kis taraf? kis shart pe? Phir saboot ya counter-example. 🎯

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✏️ Solved Examples

Example 1 (Easy): “Do odd numbers ka jod 4 ka multiple hai.”

3 + 5 = 8 ✅, 3 + 7 = 10 ❌ → kabhi-kabhi.

Check: (4p + 1) + (4q + 3) = 4(p + q + 1) ✅, (4p + 1) + (4q + 1) = 4(p + q) + 2 ❌: dono case mile ✅

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Example 2 (Medium): “9 aur 6 se bate number 54 se batega.”

LCM(9, 6) = 18. 18 ✅ dono se bata, par 54 se nahi → kabhi-kabhi.

Check: 18 ÷ 9 = 2, 18 ÷ 6 = 3, 18 ÷ 54 poora nahi ✅

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Example 3 (NCERT style): “Ek multiple of 4 aur ek multiple of 6 ka jod 2 ka multiple.”

4a + 6b = 2(2a + 3b) → hamesha.

Check: 8 + 18 = 26 = 2 × 13 ✅ (par 4 ka hamesha nahi: 4 + 6 = 10)

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Ritu ka daava sudhra: “Jo number 4 aur 6 se bate, woh 12 (LCM) se batega.” Ab hamesha sach! Sameer: “Ek counter-example ne galat daava bhi pakda aur sahi daava bhi dikhaya.” 🤝

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🏠 Daave Parakhna Roz Ki Zindagi Mein

📺 Ad ka daava: “Har customer khush!” Ek naakhush customer = daava galat.
🔬 Science: naya theory tab tak sach jab tak koi prayog (experiment) usse galat na kare.
⚖️ Court: “hamesha wahan tha” ka daava ek CCTV clip tod deti hai.
🚌 Bus timing: 12, 24 minute pe bus? 4 aur 6 minute wali dono buses saath kab: LCM 12 pe.
🛒 Packets: 6 ke packet aur 9 ke packet: dono se barabar kharidne ka chhota number LCM 18.

Agli baar koi “hamesha” bole, ek counter-example dhundhna: yeh maths wala super-power hai! 🦸

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Hamesha?	✅ Algebra / chitra	Udaharan kaafi nahi
Kabhi-kabhi?	✅ Ek haan + ek na	Dono dikhao
Kabhi nahi?	✅ Parity / tark	Contradiction
Do numbers se bhaag	✅ LCM se pakka	Guna se nahi
Ulta daava	❌ Alag se check	Disha badli

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❌→✅ Worked Mistake

Sawaal: “Jo number 6 aur 8 se bate, woh 48 se batega.” Hamesha?

❌ Galat: “48, 96, 144: sab bate. Hamesha!” (Sirf udaharan, aur guna wala niyam.)

✅ Sahi: LCM(6, 8) = 24. 24 dono se bata, 48 se nahi → kabhi-kabhi. Pakka sirf 24 se. Check: 24 ÷ 6 = 4, 24 ÷ 8 = 3 ✅

Doosri common galti: “a, M + N ko baante → M aur N dono ko baante.” Ulta niyam: 10 | 17 + 13 par 10 ∤ 17.

🧠 Rule of thumb: Hazaar udaharan saboot nahi; ek counter-example faisla.

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📝 10 MCQs

Q1. 6 dono M aur N ko baante. M − N?

(a) Hamesha 6 se

(b) Kabhi-kabhi

(c) Kabhi nahi

(d) Sirf M > N

Solution dekho

Sahi jawab: (a)

(a) sahi: 6(a − b): hamesha 6 ka multiple ✅
(b) galat: Algebra ne hamesha dikha diya.
(c) galat: Hamesha bat-ta hai, kabhi nahi galat.
(d) galat: Negative bhi 6 ka multiple hota.

Q2. Jod 8 ka multiple → dono 8 ke?

(a) Hamesha

(b) Kabhi-kabhi

(c) Kabhi nahi

(d) Sirf even

Solution dekho

Sahi jawab: (b)

(a) galat: 5 + 3 = 8, dono 8 ke nahi.
(b) sahi: 16 = 8 + 8 haan, 5 + 3 na ✅
(c) galat: 8 + 8 = 16 to haan wala case hai.
(d) galat: Even ka koi khaas niyam nahi.

Q3. 7 se bate → 21 se?

(a) Hamesha

(b) Kabhi-kabhi

(c) Kabhi nahi

(d) Sirf odd

Solution dekho

Sahi jawab: (b)

(a) galat: 14 sirf 7 se bata, 21 se nahi.
(b) sahi: 42 mein haan, 14 mein na ✅
(c) galat: 21 khud dono se bata hai.
(d) galat: 35 odd par 21 se nahi bata.

Q4. 4 aur 9 se bate → pakka kis se?

(a) 36

(b) 13

(c) 18

(d) 72

Solution dekho

Sahi jawab: (a)

(a) sahi: LCM(4, 9) = 36, common factor nahi ✅
(b) galat: 13 to jod hai, LCM nahi.
(c) galat: 18 mein 4 nahi bat-ta.
(d) galat: 72 bahut bada, 36 bhi chal jaata.

Q5. 4 aur 6 se bate → pakka kis se?

(a) 24

(b) 12

(c) 10

(d) 2 hi

Solution dekho

Sahi jawab: (b)

(a) galat: 12 dono se bata par 24 se nahi.
(b) sahi: LCM(4, 6) = 12, yahi pakka ✅
(c) galat: 10 dono ka jod hai, LCM nahi.
(d) galat: 12 bhi pakka hai, sirf 2 nahi.

Q6. Teen lagaataar odd ka jod?

(a) Hamesha 6 ka multiple

(b) Hamesha 3 ka multiple

(c) Hamesha even

(d) Kabhi 3 ka nahi

Solution dekho

Sahi jawab: (b)

(a) galat: 6n + 3 odd, 6 ka multiple nahi.
(b) sahi: 6n + 3 = 3(2n + 1) hai ✅
(c) galat: 6n + 3 hamesha odd hota hai.
(d) galat: Hamesha 3 ka multiple hota hai.

Q7. Even + odd = 10 ka multiple?

(a) Hamesha

(b) Kabhi-kabhi

(c) Kabhi nahi

(d) Sirf bade

Solution dekho

Sahi jawab: (c)

(a) galat: Jod odd hai, 10 ka nahi.
(b) galat: Ek bhi haan wala case nahi milta.
(c) sahi: Odd jod, 10 ke multiples even ✅
(d) galat: Size se parity nahi badalti.

Q8. 12 se bate number ke liye kaun galat?

(a) 3 se bata

(b) 8 se bata

(c) 4 se bata

(d) 6 se bata

Solution dekho

Sahi jawab: (b)

(a) galat: 3, 12 ka factor hai: sach.
(b) sahi: 12 khud 8 se nahi bata ✅
(c) galat: 4, 12 ka factor hai: sach.
(d) galat: 6, 12 ka factor hai: sach.

Q9. “Hamesha” saabit karne ka sahi tareeka?

(a) 10 udaharan

(b) Algebra se

(c) Ek udaharan

(d) Teacher se poochho

Solution dekho

Sahi jawab: (b)

(a) galat: Udaharan saboot nahi, sirf ishaara.
(b) sahi: Algebra har value ek saath ✅
(c) galat: Ek udaharan se kuch saabit nahi.
(d) galat: Tark khud se saabit karna hai.

Q10. Kaunsa daava “kabhi nahi”?

(a) 3 odd ka jod even

(b) 2 even ka jod even

(c) Odd × odd odd

(d) 4 ka multiple even

Solution dekho

Sahi jawab: (a)

(a) sahi: Teen odd ka jod hamesha odd ✅
(b) galat: Yeh to hamesha sach hai.
(c) galat: Yeh bhi hamesha sach hai.
(d) galat: 4k = 2(2k): hamesha even.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Counter-example kya hai?

Ek aisa udaharan jo daave ki shart poori kare par nateeja na de. “Hamesha” ko todne ke liye ek kaafi.

2. LCM wala niyam kyun?

Number mein dono ke prime factors hone chahiye; common factor dobara nahi chahiye. Wahi LCM.

3. Agar dono number co-prime hon?

To LCM = guna. 5 aur 8 → 40. Isliye 2 aur 3 se bate → 6 se bhi (6 ka divisibility test).

4. Chitra wala saboot kaise?

Multiples ko k-k ki panktiyon mein socho. Poori panktiyaan jodo → poori panktiyaan. Adhoori milke poori bhi ban sakti.

5. Kabhi-kabhi mein kitne udaharan?

Kam se kam do: ek jahan sach, ek jahan jhooth.

6. Yeh aage kahan?

Har saboot mein: Class 9-10 mein number theory, aur geometry ke theorems.

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy “5 do numbers ko baante to unke jod ko bhi.” ?

Solution dekho

Hamesha: 5a + 5b = 5(a + b). Check: 15 + 25 = 40 ✅

Q2. Easy “3 se bate → 9 se bhi.”

Solution dekho

Kabhi-kabhi: 18 ✅, 12 ❌. Check ✅

Q3. Easy “9 se bate → 3 se bhi.”

Solution dekho

Hamesha: 9m = 3(3m). Check: 27 ✅

Q4. Easy “Do even ka jod odd.”

Solution dekho

Kabhi nahi: 2a + 2b = 2(a + b). Check ✅

Q5. Medium “Do even ka jod 3 ka multiple.”

Solution dekho

Kabhi-kabhi: 2 + 4 = 6 ✅, 2 + 6 = 8 ❌. Check ✅

Q6. Medium “10 se nahi bata → 5 se bhi nahi.”

Solution dekho

Kabhi-kabhi: 15 (10 se nahi, 5 se haan); 7 (dono se nahi). Check ✅

Q7. Medium “Multiple of 4 + multiple of 8 = multiple of 4.”

Solution dekho

Hamesha: 4a + 8b = 4(a + 2b). Check: 12 + 16 = 28 ✅

Q8. Medium “Multiple of 4 + multiple of 2 = multiple of 8.”

Solution dekho

Kabhi-kabhi: 4 + 4 = 8 ✅, 4 + 2 = 6 ❌. Check ✅

Q9. NCERT style “3 aur 10 se bate → 30 se.”

Solution dekho

Hamesha: co-prime, LCM 30. Check: 60 ✅

Q10. NCERT style “Teen lagaataar numbers ka guna 6 ka multiple.”

Solution dekho

Hamesha: ek even, ek 3 ka multiple. Check: 4 × 5 × 6 = 120 ✅

Q11. NCERT style “5(3b + 2) − 3(5b − 1) 13 ka multiple.”

Solution dekho

15b + 10 − 15b + 3 = 13 → hamesha. Check: b = 1 → 25 − 12 = 13 ✅

Q12. NCERT style “Do lagaataar even numbers ka guna 8 ka multiple.”

Solution dekho

Hamesha: 2n(2n + 2) = 4n(n + 1), n(n + 1) even. Check: 6 × 8 = 48 ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. “a, b dono 7 se nahi bate → a + b bhi 7 se nahi.”

Solution dekho

Kabhi-kabhi: 1 + 2 = 3 (7 se nahi, daava sach), par 3 + 4 = 7 (7 se bata, daava toota). Check ✅

Q2. Kaunsa sabse chhota number 2, 3, 4, 5, 6 sab se bat-ta?

Solution dekho

LCM = 60. Check: 60 ÷ 4 = 15, ÷ 6 = 10 ✅

Q3. “Paanch lagaataar numbers ka guna 120 ka multiple.”

Solution dekho

Hamesha: 8, 3, 5 sab factors milte. Check: 1 × 2 × 3 × 4 × 5 = 120 ✅

Q4. “n³ − n hamesha 6 ka multiple.”

Solution dekho

Hamesha: (n − 1)n(n + 1) teen lagaataar. Check: n = 4 → 60 ✅

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🏆 Olympiad Corner

Q1. 12 aur 15 dono se bate sabse chhota 4-digit number?

Solution dekho

LCM 60 → 1020.

Q2. 1 se 100 mein 4 aur 6 dono se bate kitne?

Solution dekho

12 ke multiples → 8.

Q3. n² + n + 1 kabhi even?

Solution dekho

Kabhi nahi: n(n + 1) even + 1.

Q4. Do numbers ka jod 9 ka multiple aur ek 9 ka multiple. Doosra?

Solution dekho

Hamesha 9 ka multiple (farak).

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💗 CharuMam Motivation

Sameer ne Ritu ko haraaya nahi, dono ne milke daava sudhaara. Maths mein counter-example dushman nahi, dost hai! 🤝

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🟢 Master Recap

💡 Chaabi: Hamesha saabit karne ke liye algebra (ya chitra) chahiye, udaharan kaafi nahi. Kabhi nahi ke liye bhi tark chahiye. Kabhi-kabhi ke liye bas ek “haan” aur ek “na” udaharan. Aur “hamesha” todne ke liye ek counter-example kaafi.
Hamesha: algebra/chitra saboot.
Kabhi-kabhi: ek haan, ek na.
Kabhi nahi: parity ya tark.
a | M, a | N → a | M ± N (ulta nahi).
Multiples aur factors ka niyam.
Do numbers se bhaag → LCM se pakka.

— charumam.com 🎯

👉 Aage Kya?

Bhaag ke baad jo bachta hai, uska bhi khel hai! Agla post: Post 4 — Remainder Ka Khel.

← Pichla: Post 2 — Even-Odd Ka Khel: Parity 📚 Chapter 5 ke saare posts Agla: Post 4 — Remainder Ka Khel →

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

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Class Ka Debate: “Yeh Hamesha Sach Hai!”

📖 Concept: Teen Dibbe, Teen Tarah Ke Saboot

Step 1: Daava 1: 6 dono ko baante to jod ko bhi

Step 2: Daava 2: jod 10 ka multiple → dono number bhi?

Step 3: Daava 3 aur 4: multiples aur factors

Step 4: Daava 5: do numbers se bate → kis se pakka?

Step 5: Daava 6: kabhi nahi ka saboot

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Udaharan Kyun Kaafi Nahi? (Ek Kaala Hans Ki Kahani)

✏️ Solved Examples

🏠 Daave Parakhna 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?

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