Consecutive Numbers Ka Jod | Class 8 Maths Book 1 Ch 5

📘 Class 8 · Ganita Prakash Book 1Ch 5 · Number PlayPost 2 / 7⏱ 25–30 min

Even-Odd Ka Khel: Parity ⚖️

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:

Even number ki pehchaan?
Odd + odd = ?
2n kaisa number hai (n integer)?

Answers dekho

2 se poora bhaag (aakhri digit 0, 2, 4, 6, 8)
Even
Hamesha even

Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 1 — Consecutive Numbers Ka Jod dekh lijiye.

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🎬 Kahani: Kabir Ka Jaadu: “Main Pehle Se Jaanta Hoon!”

Kabir ne Meera se kaha: “Koi bhi chaar lagaataar numbers lo, beech mein + ya − jaise chaaho lagao. Main bina dekhe bata dunga: jawab even hoga!” 🎩

Meera ne 11, 12, 13, 14 liye: 11 − 12 + 13 − 14 = −2. Even! Phir 11 + 12 − 13 + 14 = 24. Even! Aath tareeke, aathon even.

Kya Kabir ke paas jaadu hai? Nahi, uske paas ek niyam hai. Aaj woh niyam aur uska “kyun” dono samjhenge. ✨

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📖 Concept: Parity Sirf Odd Numbers Ki Ginti Pe Tikti Hai

💡 Is post ki ek badi baat: Jod ya ghataav ka even/odd hona (parity) sirf is baat pe tikta hai ki usmein kitne odd numbers hain: even ginti → even, odd ginti → odd. + ko − karne se parity nahi badalti.

Step 1: Teen basic niyam (aur kyun)

Operation	Parity	Kyun (jodi soch)
even ± even	even	dono ki jodiyaan poori
odd ± odd	even	dono ke akele milke jodi
even ± odd	odd	ek akela bacha

Algebra se: (2a + 1) + (2b + 1) = 2(a + b + 1): even ✅

Step 2: Explanation 1: sign badlo, farak dekho

a + b − c − d aur a − b − c − d ka farak = 2b: hamesha even!

Do numbers jinka farak even ho, unki parity same. Har sign badalne pe farak 2 × (woh number) → parity same. Aath ke aath expression ek hi parity!

WHY kaam karta? +b se −b jaane mein b do baar hata: 2b. “Do baar” = even.

Step 3: Explanation 2: odd numbers gino

11, 12, 13, 14 mein odd: 11, 13 → do → har expression even.

Chaar lagaataar mein hamesha 2 odd → hamesha even. Paanch lagaataar (odd se shuru, jaise 7–11) mein 3 odd → hamesha odd!

Kab kaunsa? Explanation 2 numbers dikhe to sabse tez. Explanation 1 tab jab letters (a, b, c) hon aur saboot chahiye.

Step 4: Breaking even: bina hisaab ke pehchaano

57 + 29 → odd + odd → even
846 − 375 → even − odd → odd
5 × 213 × 7 → koi factor even nahi → odd
91 × 44 → ek factor even → even

Guna ka niyam: ek bhi even factor → even. Kyunki 2 factor ban gaya. Sab odd → odd.

Step 5: Algebra mein “hamesha even?”

6a + 4b = 2(3a + 2b) → hamesha
5k − 3k = 2k → hamesha
3m + n → m = 1, n = 1: 4 even; m = 1, n = 2: 5 odd → kabhi-kabhi
y² + 3 → y = 1: 4; y = 2: 7 → kabhi-kabhi
y² + y = y(y + 1) → do lagaataar mein ek even → hamesha!

Step 6: 4 wale jode: even ke do prakaar

Even numbers 4 se bhaag pe: 0 bache (4p) ya 2 bache (4p + 2).

4p + 4q = 4(p + q) → 4 ka multiple
(4p + 2) + (4q + 2) = 4(p + q + 1) → 4 ka multiple! (14 + 30 = 44)
4p + (4q + 2) = 4(p + q) + 2 → nahi (20 + 10 = 30)

Bilkul odd/even jaisa: “2 wale” do mile to poora 4 ban gaya.

🪜 Atak Gaye? Aise Socho

Har number ke aage likho: E ya O.
Jod/ghataav: O gino. Even ginti → even jawab.
Guna: ek bhi E → even.
Letters ho: 2 bahar nikaal sakte ho? To hamesha even.
Shak ho: ek even, ek odd value daal ke try karo.

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🔐 Deep Concept: Parity Ek “Chhota Hisaab” Hai (Sirf 0 Aur 1 Ki Duniya)

Socho har number ko sirf uske “bache hue” se yaad rakho: even = 0, odd = 1. Ab hisaab:

0 + 0 = 0, 1 + 1 = 2 → wapas 0, 0 + 1 = 1. Aur ghataav mein bhi wahi: 1 − 1 = 0, 0 − 1 = −1 → odd (1).

Is duniya mein + aur − ek hi hain! Kyunki +1 aur −1 dono “odd” hain. Isliye Kabir ka sign badalna kuch nahi badalta. 🎯

Token model: + aur − tokens ki ek jodi zero banaati. Sign badalna = b tokens ki jagah b ulte tokens = 2b ka badlaav = poori jodiyaan.

Unifying idea: bade numbers ki jagah sirf “bachi hui” cheez dekho. Yahi soch Post 4 — Remainder Ka Khel (remainder) aur Post 5 — Divisibility Rule: 9, 3 Aur 11 (divisibility rules) ki neev hai.

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✏️ Solved Examples

Example 1 (Easy): 1 ± 2 ± 3 ± 4 ± 5 ± 6 se 0 bana sakte?

Odd: 1, 3, 5 → teen → har expression odd. 0 even hai → kabhi nahi.

Check: 1 + 2 + 3 + 4 + 5 + 6 = 21 odd; ek sign badlo, 21 − 2 × (koi number) bhi odd ✅

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Example 2 (Medium): 1 ± 2 ± … ± 7 se 0?

Odd: 1, 3, 5, 7 → chaar → even, mumkin! Kul 28, to + wale ka jod 14 chahiye: 1 + 6 + 7.

Check: 1 − 2 − 3 − 4 − 5 + 6 + 7 = 0 ✅

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Example 3 (NCERT style): 4m + 6n hamesha 4 ka multiple?

4m + 6n = 4m + 4n + 2n → 2n pe nirbhar. n = 1: 10 ❌; n = 2: 4m + 12 ✅ → kabhi-kabhi (jab n even).

Check: m = 1, n = 1 → 10, 4 se nahi ✅

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Kabir ka raaz: chaar lagaataar numbers mein hamesha do odd, aur sign badalne se parity nahi badalti. Meera ne kaha: “Ab main paanch numbers se tumhe chakma dungi!” Kabir: “Pehle odd gin lena!” 😄

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🏠 Parity Roz Ki Zindagi Mein

🚪 Light switch: har dabaane pe on/off badalta: even baar dabao to wapas wahi. Parity!
🏠 Ghar ke number: sadak ke ek taraf even, doosri taraf odd: dakiya ek nazar mein side pehchaanta.
🎲 Do paase: jod even aane ke liye dono even ya dono odd chahiye.
🪑 Musical chairs / jodi game: odd bachche → ek akela bachega, pakka.
💻 Computer: data bhejte waqt “parity bit” se galti pakdi jaati: 1s ki ginti even honi chahiye.

Computer wala kamaal: agar ek bhi bit galti se badla to ginti odd ho jaati aur galti pakdi jaati. Yahi aaj ka niyam!

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🧭 Kab Use Karein, Kab Nahi

Situation	Kya karein	Kyun
Numbers diye	✅ Odd gino	Even ginti → even
Letters diye	✅ 2 bahar nikaalo	Ya sign-farak 2b
Guna	✅ Ek even factor kaafi	Sab odd → odd
4 ka multiple?	✅ 4p ya 4p + 2 dekho	Do “2 wale” = 4
Kabhi-kabhi?	✅ Ek haan, ek na udaharan	Dono dikhao

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❌→✅ Worked Mistake

Sawaal: 2 − 5 − 9 + 4 − 7 even ya odd?

❌ Galat: “Minus bahut hain, to jawab negative, to odd.” Sign ka parity se koi lena-dena nahi!

✅ Sahi: odd: 5, 9, 7 → teen → odd. Check: 2 − 5 − 9 + 4 − 7 = −15 → odd ✅

Doosri common galti: “Negative number even nahi ho sakta.” Galat: −8 = 2 × (−4) → even.

🧠 Rule of thumb: Sign bhool jao, odd gino: even ginti = even jawab.

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📝 10 MCQs

Q1. 39 + 85 even ya odd?

(a) Even

(b) Odd

(c) Dono

(d) Pata nahi

Solution dekho

Sahi jawab: (a)

(a) sahi: Odd + odd = even, jodi poori ✅
(b) galat: Do odd mil ke even bante hain.
(c) galat: Ek number dono nahi ho sakta.
(d) galat: Bina jode bhi pata chal jaata.

Q2. 7 × 15 × 9?

(a) Even

(b) Odd

(c) Zero

(d) Negative

Solution dekho

Sahi jawab: (b)

(a) galat: Koi factor even nahi, to even nahi.
(b) sahi: Sab factor odd → odd ✅
(c) galat: Guna zero nahi, koi factor 0 nahi.
(d) galat: Sab positive hai, negative nahi.

Q3. a + b − c aur a − b − c ka farak?

(a) 2a

(b) 2b

(c) 2c

(d) 0

Solution dekho

Sahi jawab: (b)

(a) galat: a ka sign nahi badla, farak 2b.
(b) sahi: +b se −b hua: farak 2b ✅
(c) galat: c dono mein same sign mein hai.
(d) galat: Farak zero nahi, b badla hai.

Q4. Kaunsa hamesha even (n integer)?

(a) 3n + 1

(b) n² + n

(c) n + 5

(d) 5n

Solution dekho

Sahi jawab: (b)

(a) galat: n = 1 pe 4, n = 2 pe 7: kabhi-kabhi.
(b) sahi: n(n + 1): lagaataar mein ek even ✅
(c) galat: n = 2 pe 7 odd aata hai.
(d) galat: n = 1 pe 5 odd aata hai.

Q5. 5 lagaataar (odd se shuru) ka ± expression?

(a) Hamesha even

(b) Hamesha odd

(c) Kabhi-kabhi

(d) Zero

Solution dekho

Sahi jawab: (b)

(a) galat: Teen odd hain, to even nahi.
(b) sahi: Teen odd → hamesha odd ✅
(c) galat: Parity kabhi nahi badalti, fixed hai.
(d) galat: Odd hai, to zero nahi ho sakta.

Q6. −14 kaisa number?

(a) Odd

(b) Even

(c) Na even na odd

(d) Prime

Solution dekho

Sahi jawab: (b)

(a) galat: 2 se poora bhaag hota, odd nahi.
(b) sahi: −14 = 2 × (−7) → even ✅
(c) galat: Har integer even ya odd hota hai.
(d) galat: Prime positive hote, aur 2 factor hai.

Q7. (4p + 2) + (4q + 2) = ?

(a) 4(p + q) + 2

(b) 4(p + q + 1)

(c) 2(p + q)

(d) 4pq

Solution dekho

Sahi jawab: (b)

(a) galat: 2 + 2 = 4, bacha nahi.
(b) sahi: 2 + 2 = 4 → 4(p + q + 1) ✅
(c) galat: Galat factor nikaala gaya hai.
(d) galat: Yeh jod hai, guna nahi.

Q8. 1 ± 2 ± 3 ± 4 se 0?

(a) Nahi, jawab odd aata

(b) Haan, sirf ek tareeke se

(c) Haan, teen tareekon se

(d) Nahi, hamesha negative

Solution dekho

Sahi jawab: (b)

(a) galat: Odd do hain → even, to mumkin.
(b) sahi: 1 − 2 − 3 + 4 = 0, bas yahi ✅
(c) galat: Minus wale ka jod 5: sirf 2 + 3.
(d) galat: 1 + 2 + 3 + 4 = 10 positive bhi.

Q9. 3m + n kab even?

(a) Hamesha

(b) m, n dono odd ya dono even

(c) Kabhi nahi

(d) Sirf n = 0

Solution dekho

Sahi jawab: (b)

(a) galat: m = 1, n = 2 pe 5 odd hai.
(b) sahi: 3m ki parity m jaisi: same parity ✅
(c) galat: m = n = 1 pe 4 even hai.
(d) galat: n = 2, m = 2 pe bhi even.

Q10. Light switch 25 baar dabaaya (shuru off). Ab?

(a) Off

(b) On

(c) Pata nahi

(d) Toot gaya

Solution dekho

Sahi jawab: (b)

(a) galat: Odd baar dabaane pe badal jaata.
(b) sahi: 25 odd hai → ulta → on ✅
(c) galat: Parity se pakka pata hai.
(d) galat: Toota nahi, sirf parity ka sawaal.

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❓ FAQs (Bachchon Ke Asli Sawaal)

1. Parity ka matlab?

Number even hai ya odd: bas itni jaankari. Parity = “jodi poori ya ek bacha”.

2. 0 even hai?

Haan: 0 = 2 × 0. Even aur odd ke beech bilkul fit (…, −1, 0, 1…).

3. Koi bhi 4 numbers (lagaataar nahi) mein bhi sign se parity same?

Haan! Explanation 1 kisi bhi numbers pe chalta: farak hamesha 2b. Bas kaunsi parity, woh odd ki ginti batayegi.

4. Guna mein sign badalne ka?

Guna ki parity sign se nahi badalti: (−3) × 5 = −15 odd.

5. Bhaag mein parity?

Bhaag ka koi seedha niyam nahi: 12 ÷ 2 = 6 even, 10 ÷ 2 = 5 odd.

6. Yeh kahan kaam aata?

Paheliyon (chessboard, domino), computer error check, aur saboot mein “kabhi nahi” dikhaane ke liye.

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🏋️ Practice Zone (12 Sawaal)

Q1. Easy 78 − 43 even ya odd?

Solution dekho

even − odd → odd. Check: 35 ✅

Q2. Easy 12 × 37 × 5?

Solution dekho

12 even → even. Check: 2220 ✅

Q3. Easy 2a + 8b hamesha even?

Solution dekho

Haan: 2(a + 4b). Check: a = 1, b = 1 → 10 ✅

Q4. Easy 9 − 4 + 7 − 6 + 1?

Solution dekho

Odd: 9, 7, 1 → teen → odd. Check: 7 ✅

Q5. Medium x² + 5 kab even?

Solution dekho

Jab x odd. Check: x = 3 → 14 even; x = 2 → 9 odd ✅

Q6. Medium 1 ± 2 ± … ± 5 se 1 bana sakte?

Solution dekho

Teen odd → odd jawab, 1 bhi odd → mumkin: 1 + 2 − 3 − 4 + 5 = 1. Check ✅

Q7. Medium 22 + 38 kya 4 ka multiple?

Solution dekho

Dono 4p + 2 type → haan: 60. Check: 60 ÷ 4 = 15 ✅

Q8. Medium 6m − 3n hamesha even?

Solution dekho

Nahi: n odd ho to odd. Check: m = 1, n = 1 → 3 ✅

Q9. NCERT style Kya 1 ± 2 ± … ± 10 se 0 ban sakta?

Solution dekho

Kul 55 odd (paanch odd numbers) → nahi. Check ✅

Q10. NCERT style k(k + 1)(k + 2) hamesha even?

Solution dekho

Haan: teen lagaataar mein kam se kam ek even. Check: 3 × 4 × 5 = 60 ✅

Q11. NCERT style Do numbers ka jod aur farak: kya ek even, ek odd ho sakte?

Solution dekho

Nahi: (a + b) − (a − b) = 2b → same parity. Check: 9 + 4 = 13, 9 − 4 = 5 ✅

Q12. NCERT style Teen odd numbers ka jod 50 ho sakta?

Solution dekho

Nahi: teen odd → odd. Check ✅

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🧠 HOTS (Sochne Wale Sawaal)

Q1. Chessboard ke do aamne-saamne kone kaat do. Kya 31 dominoes (har ek 2 khaane) se baaki 62 khaane dhak sakte?

Solution dekho

Nahi: kate kone ek rang ke → 30 ek rang, 32 doosre; har domino ek-ek rang dhakta. Check: rang ki ginti ✅

Q2. 7 coins, sab head up. Har chaal mein theek 2 palto. Sab tail?

Solution dekho

Kabhi nahi: tails ki ginti hamesha even, 7 odd. Check ✅

Q3. a, b, c integers. a + b, b + c, c + a teeno odd ho sakte?

Solution dekho

Nahi: teeno ka jod 2(a + b + c) even, par teen odd ka jod odd. Check ✅

Q4. 1 ± 2 ± … ± n se 0: kin n ke liye?

Solution dekho

Jab 1 se n ka jod even: n = 3, 4, 7, 8, 11, 12… Check: n = 3 → 1 + 2 − 3 = 0 ✅

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🏆 Olympiad Corner

Q1. 1 se 99 tak ke saare numbers ka ± expression: parity?

Solution dekho

50 odd → even.

Q2. Ek party mein haath milaane: odd baar haath milaane wale log kitne ho sakte?

Solution dekho

Hamesha even (kul haath milaane × 2 even).

Q3. n² − n hamesha kiska multiple?

Solution dekho

2: (n − 1)n lagaataar.

Q4. Do even numbers jo 4 ke multiple nahi, unka guna kis ka multiple?

Solution dekho

4 (2 × 2 factors), par 8 ka nahi.

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💗 CharuMam Motivation

Kabir ka “jaadu” ek chhota sa niyam tha: odd gino. Maths ki taakat yahi: ek baar saboot mil gaya to anginat (infinite) cases bina gine sach! 💪

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🟢 Master Recap

💡 Chaabi: Jod ya ghataav ka even/odd hona (parity) sirf is baat pe tikta hai ki usmein kitne odd numbers hain: even ginti → even, odd ginti → odd. + ko − karne se parity nahi badalti.
Parity = even ya odd.
Jod/ghataav: odd numbers ki ginti dekho.
Guna: ek even factor → even.
Sign badalne se farak 2b → parity same.
Even ke do prakaar: 4p aur 4p + 2.
Kabhi-kabhi: ek haan, ek na udaharan do.

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👉 Aage Kya?

Kaun sa daava hamesha sach, kaun kabhi-kabhi, kaun kabhi nahi? Agla post: Post 3 — Hamesha, Kabhi-Kabhi Ya Kabhi Nahi?.

← Pichla: Post 1 — Consecutive Numbers Ka Jod 📚 Chapter 5 ke saare posts Agla: Post 3 — Hamesha, Kabhi-Kabhi Ya Kabhi Nahi? →

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

✅ Pehle Yeh Aata Hai?

🎬 Kahani: Kabir Ka Jaadu: “Main Pehle Se Jaanta Hoon!”

📖 Concept: Parity Sirf Odd Numbers Ki Ginti Pe Tikti Hai

Step 1: Teen basic niyam (aur kyun)

Step 2: Explanation 1: sign badlo, farak dekho

Step 3: Explanation 2: odd numbers gino

Step 4: Breaking even: bina hisaab ke pehchaano

Step 5: Algebra mein “hamesha even?”

Step 6: 4 wale jode: even ke do prakaar

🪜 Atak Gaye? Aise Socho

🔐 Deep Concept: Parity Ek “Chhota Hisaab” Hai (Sirf 0 Aur 1 Ki Duniya)

✏️ Solved Examples

🏠 Parity 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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