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Class 7 Maths Part 2 Chapter 3: Finding Common Ground (Hinglish Notes)

Ye chapter 5 lines mein

Finding Common Ground HCF aur LCM ke baare mein hai. Kuch numbers ka HCF unke common factors mein sabse bada hota hai, aur LCM unke common multiples mein sabse chhota. Dono prime factorisation se aasaani se milte hain: HCF har common prime ko utni baar leta hai jitni sabse kam baar wo aata hai, aur LCM har prime ko utni baar jitni sabse zyaada baar. Do numbers ke liye HCF × LCM unke guna ke barabar hota hai.

Part 2 Chapter 3 ka mind map: common factors aur HCF, prime factorisation, primes se HCF, common multiples aur LCM, aur patterns aur aasaan tareeka
Poora chapter ek page pe

Ye chapter kis baare mein hai?

Rozmarra ke kai sawaal kuch aisa poochhte hain jo do numbers mein common ho. Kamre mein poori tarah fit hone waali sabse badi square tile kaunsi hai? Do ghatnaayein dobara ek hi din kab hongi? Pehle sawaal ke liye highest common factor (HCF) chahiye aur doosre ke liye lowest common multiple (LCM). Class 6 mein tumne inhe list banaakar nikaala tha; ye chapter prime factorisation se inhe jaldi aur bharose se nikaalta hai, aur phir patterns aur ek chhota tareeka dhoondhta hai.

Common factors aur HCF

Sameeksha 12 ft by 16 ft ke farsh ko barabar square tiles se dhakna chahti hai, kam se kam tiles mein, aur tile ki side poore feet mein. Tile ki side 12 aur 16 dono ko bhaag de, isliye wo common factor hogi: 1, 2 ya 4. Sabse badi, 4 ft, mein sabse kam tiles lagti hain, sirf 3 × 4 = 12.

12 ft by 16 ft ka kamra 1 ft, 2 ft aur 4 ft tiles se: 192, 48 aur 12 tiles
12 aur 16 ke common factors

Do ya zyaada numbers ka HCF unke common factors mein sabse bada hota hai; ise GCD (greatest common divisor) bhi kehte hain. 84 kg aur 108 kg chawal ko barabar bags mein bharne pe 12 kg ke bags se sabse kam bags lagte hain, kyunki 12 HCF hai. Jump Jackpot game mein 0 se shuru hokar dono khazaanon pe utarne waali sabse lambi jump bhi HCF hai.

Prime factorisation

Bade numbers ke factors ki list banaana dheema hai aur galti aasaani se ho jaati hai. Iski jagah har number ko primes mein todo. Division method ek baar mein ek prime se bhaag deta hai aur har bhaagfal neeche likhta hai, jab tak 1 na bache. Factor tree number ko kinhi do factors mein todta hai aur todta rehta hai. Dono tarah primes wahi aate hain, bas kram alag ho sakta hai.

1200 ka division method aur 90 ka factor tree
1200 = 2 × 2 × 2 × 2 × 3 × 5 × 5

Kisi number ka har factor uske prime factorisation ka ek hissa hai. Kyunki 840 = 2 × 2 × 2 × 3 × 5 × 7, hissa 2 × 2 × 7 = 28 ek factor hai, aur baaki 2 × 3 × 5 = 30 use guna karke 840 banaata hai. Par 3 × 3 × 3 factor nahi hai, kyunki 840 mein sirf ek 3 hai. Har sambhav hissa lene se saare factors mil jaate hain.

225 = 3 × 3 × 5 × 5 se uske nau factors banaane waala grid
225 ke factors

Anshu ne andaaza lagaaya ki bade number ka prime factorisation hamesha lamba hota hai. Aisa andaaza conjecture hai, aur ek counterexample use galat saabit kar deta hai: 96 = 2 × 2 × 2 × 2 × 2 × 3 mein chhe primes hain, par bade 121 = 11 × 11 mein sirf do.

Prime factors se HCF

Common factor har factorisation ka hissa hona chahiye, isliye HCF sabse bada saajha hissa hai: har common prime ko utni baar lo jitni sabse kam baar wo aata hai. 30 = 2 × 3 × 5 aur 72 = 2 × 2 × 2 × 3 × 3 ka HCF 2 × 3 = 6 hai. Agar koi prime saajha na ho, jaise 96 aur 275 mein, to HCF 1 hai. Aur padho HCF kaise nikalein? mein.

72 aur 120 ke prime factors 2s, 3s aur 5s mein bante hue, HCF sabse kam aur LCM sabse zyaada leta hai
HCF 24 aur LCM 360

Common multiples aur LCM

Anshu ke toran 6 cm ki pattiyon se bante hain aur Guna ke 8 cm ki pattiyon se. Barabar lambe torans ki lambai 6 aur 8 ka common multiple honi chahiye, jaise 24 ya 48 cm, aur sabse chhoti 24 cm hai. Sabse bada common multiple hota hi nahi, kyunki kisi ko dugna karne se ek aur mil jaata hai. Isi tarah agar ek dukaan har Monday gajak baantti hai aur Kabamai har 10 din mein aati hai, to use agli baar gajak 7 aur 10 ke LCM, yaani 70 din baad milegi.

6 cm aur 8 cm ki pattiyon ki kataarein 24 cm aur 48 cm pe saath khatm hoti hain
6 aur 8 ka LCM 24

Kisi number ke multiple mein us number ke saare primes hote hain, isliye common multiple mein har number ka har prime hota hai. LCM har prime ko utni baar rakhta hai jitni sabse zyaada baar wo aata hai: 96 = 2 × 2 × 2 × 2 × 2 × 3 aur 360 = 2 × 2 × 2 × 3 × 3 × 5 ka LCM 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 1440 hai. Dekho LCM kaise nikalein?.

Patterns aur ek aasaan tareeka

Agar ek number doosre ka factor ho, jaise 6 aur 18, to chhota number HCF hai aur bada LCM. Lagaatar numbers aur co-prime numbers ka HCF 1 hai. Dono jawab ek saath paane ke liye dono numbers ko ek ladder mein common primes (ya kisi bhi common factor) se bhaag do, jab tak kuch saajha na bache.

84 aur 180 ko 2, 2 aur 3 se bhaag dekar 7 aur 15 tak pahunchne waala ladder
HCF 12 aur LCM 1260

Divisors se HCF milta hai, 2 × 2 × 3 = 12, aur divisors ko bache hue numbers 7 aur 15 se guna karne pe LCM, 1260. Do numbers ke liye HCF × LCM = numbers ka guna: 12 × 1260 = 15,120 = 84 × 180. Padho HCF aur LCM ka sambandh. Har sawaal ka jawab Part 2 Chapter 3 ke NCERT solutions mein hai.

Key terms (zaroori shabd)

TermMatlab
FactorWo number jo kisi doosre number ko poora-poora bhaag de. 4, 12 ka factor hai.
MultipleKisi number ko 1, 2, 3 waghairah se guna karke bana number. 24, 6 ka bhi multiple hai aur 8 ka bhi.
Common factorWo number jo diye gaye har number ka factor ho. 12 aur 16 ke common factors 1, 2 aur 4 hain.
HCF (Highest Common Factor)Do ya zyaada numbers ke common factors mein sabse bada. Ise GCD (Greatest Common Divisor) bhi kehte hain.
Common multipleWo number jo diye gaye har number ka multiple ho, jaise 6 aur 8 ke liye 24 aur 48.
LCM (Lowest Common Multiple)Do ya zyaada numbers ke common multiples mein sabse chhota, jise least common multiple bhi kehte hain. 6 aur 8 ka LCM 24 hai.
Prime number1 se bada aisa number jiske factors sirf 1 aur wo khud hon, jaise 2, 3, 5, 7 aur 11.
Prime factorisationNumber ko primes ke guna ke roop mein likhna, jaise 90 = 2 × 3 × 3 × 5. Primes hamesha wahi rehte hain, sirf unka kram badal sakta hai.
Co-prime numbersAise numbers jinka ek hi common factor 1 ho, jaise 8 aur 15. Inka HCF 1 hota hai.
ConjectureBina saboot ka daava. Use galat saabit karne ke liye ek counterexample, yaani ek aisa udaharan jahan wo fail ho, kaafi hai.
GeneralisationAisa kathan jo har case mein sach ho, jaise: agar ek number doosre ka factor hai, to wahi unka HCF hai.

Is chapter ke topics

Is chapter ke NCERT solutions Chapter ke saare exercise questions, step-by-step jawab ke saath.

Practice karo

  1. 18 aur 24 ka HCF hai:

    1. (a) 3
    2. (b) 6
    3. (c) 12
    4. (d) 72
    Jawab dekho

    Jawab: (b) 6

  2. 4 aur 10 ka LCM hai:

    1. (a) 2
    2. (b) 14
    3. (c) 20
    4. (d) 40
    Jawab dekho

    Jawab: (c) 20

  3. 60 ka prime factorisation kaunsa hai?

    1. (a) 6 × 10
    2. (b) 2 × 2 × 3 × 5
    3. (c) 2 × 3 × 10
    4. (d) 4 × 15
    Jawab dekho

    Jawab: (b) 2 × 2 × 3 × 5

  4. Do numbers ka HCF 4 aur LCM 48 hai. Ek number 12 hai. Doosra hai:

    1. (a) 16
    2. (b) 4
    3. (c) 48
    4. (d) 36
    Jawab dekho

    Jawab: (a) 16 HCF × LCM = guna, isliye doosra number = 4 × 48 ÷ 12 = 16.

  5. Do lagaatar numbers ka HCF hamesha hota hai:

    1. (a) 0
    2. (b) 1
    3. (c) 2
    4. (d) unka guna
    Jawab dekho

    Jawab: (b) 1

  6. Do ghantiyan har 6 minute aur har 9 minute pe bajti hain. 9:00 baje dono saath baji. Agli baar saath kab bajengi?

    1. (a) 9:15
    2. (b) 9:18
    3. (c) 9:54
    4. (d) 9:36
    Jawab dekho

    Jawab: (b) 9:18 6 aur 9 ka LCM 18 hai, isliye 18 minute baad.

Students ke sawal

Class 7 Maths Part 2 Chapter 3 ke main topics kya hain?

Common factors aur HCF; division method se prime factorisation; primes se kisi number ke saare factors nikaalna; prime factorisation se HCF; common multiples aur LCM; HCF aur LCM ke general kathan; ek aasaan common division method; aur do numbers ke liye HCF × LCM = guna ka niyam.

Kya Finding Common Ground nayi Ganita Prakash Part 2 book mein hai?

Haan. Ye Class 7 Maths ki textbook Ganita Prakash Part 2 (2026-27) ka Chapter 3 hai, pages 47 se 66 tak.

Prime factorisation se HCF aur LCM kaise nikaalte hain?

Har number ko primes ke guna mein likho. HCF ke liye sabhi numbers mein common primes ko guna karo, har ek ko sabse kam baar jitna wo aata hai. LCM ke liye har prime ko guna karo, har ek ko sabse zyaada baar jitna wo aata hai. 72 aur 120 ka HCF 24 aur LCM 360 hai.

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